Chapter 12 Resource Mastersrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812... · 1 5 ( 7) or 2 The common difference is 2. Add 2 to the third term to get the fourth term, and so

Chapter 12Resource Masters

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ISBN: 0-07-869139-7 Advanced Mathematical ConceptsChapter 12 Resource Masters

1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04

© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

Vocabulary Builder . . . . . . . . . . . . . . . . vii-ix

Lesson 12-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 509Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 511









Chapter 12 AssessmentChapter 12 Test, Form 1A . . . . . . . . . . . 537-538Chapter 12 Test, Form 1B . . . . . . . . . . . 539-540Chapter 12 Test, Form 1C . . . . . . . . . . . 541-542Chapter 12 Test, Form 2A . . . . . . . . . . . 543-544Chapter 12 Test, Form 2B . . . . . . . . . . . 545-546Chapter 12 Test, Form 2C . . . . . . . . . . . 547-548Chapter 12 Extended Response

Assessment . . . . . . . . . . . . . . . . . . . . . . . 549Chapter 12 Mid-Chapter Test . . . . . . . . . . . . 550Chapter 12 Quizzes A & B . . . . . . . . . . . . . . 551Chapter 12 Quizzes C & D. . . . . . . . . . . . . . 552Chapter 12 SAT and ACT Practice . . . . 553-554Chapter 12 Cumulative Review . . . . . . . . . . 555Trigonometry Semester Test . . . . . . . . . 557-561Trigonometry Final Test . . . . . . . . . . . . 563-570

SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1

SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A23

Contents

© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

A Teacher’s Guide to Using theChapter 12 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 12 Resource Masters include the corematerials needed for Chapter 12. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.

All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.

Vocabulary Builder Pages vii-ix include a student study tool that presents the key vocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.

When to Use Give these pages to studentsbefore beginning Lesson 12-1. Remind them toadd definitions and examples as they completeeach lesson.

Study Guide There is one Study Guide master for each lesson.

When to Use Use these masters as reteachingactivities for students who need additional reinforcement. These pages can also be used inconjunction with the Student Edition as aninstructional tool for those students who havebeen absent.

Practice There is one master for each lesson.These problems more closely follow the structure of the Practice section of the StudentEdition exercises. These exercises are of averagedifficulty.

When to Use These provide additional practice options or may be used as homeworkfor second day teaching of the lesson.

Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they are learning. These are not written exclusively forhonors students, but are accessible for use withall levels of students.

When to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.

© Glencoe/McGraw-Hill v Advanced Mathematical Concepts

Assessment Options

The assessment section of the Chapter 12Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessments

Chapter Tests• Forms 1A, 1B, and 1C Form 1 tests contain

multiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with average-level students, and Form 1C is intended foruse with basic-level students. These tests aresimilar in format to offer comparable testingsituations.

• Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.

All of the above tests include a challengingBonus question.

• The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided for assessment.

Intermediate Assessment• A Mid-Chapter Test provides an option to

assess the first half of the chapter. It iscomposed of free-response questions.

• Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter.

Continuing Assessment• The SAT and ACT Practice offers

continuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitative-comparison, and grid-in questions. Bubble-in and grid-in answer sections are providedon the master.

• The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.

Answers• Page A1 is an answer sheet for the SAT and

ACT Practice questions that appear in theStudent Edition on page 835. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they may encounterin test taking.

• The answers for the lesson-by-lesson masters are provided as reduced pages withanswers appearing in red.

• Full-size answer keys are provided for theassessment options in this booklet.

© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts

Chapter 12 Leveled Worksheets

Glencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow.

• Study Guide masters provide worked-out examples as well as practiceproblems.

• Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords.

• Practice masters provide average-level problems for students who are moving at a regular pace.

• Enrichment masters offer students the opportunity to extend theirlearning.

primarily skillsprimarily conceptsprimarily applications

BASIC AVERAGE ADVANCED

Study Guide

Vocabulary Builder

Parent and Student Study Guide (online)

Practice

Enrichment

4

5

3

2

Five Different Options to Meet the Needs of Every Student in a Variety of Ways

1

© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

Reading to Learn MathematicsVocabulary Builder

NAME _____________________________ DATE _______________ PERIOD ________

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 12.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.

Vocabulary Term Foundon Page Definition/Description/Example

arithmetic mean

arithmetic sequence

arithmetic series

Binomial Theorem

common difference

common ratio

comparison test

convergent series

divergent series

escaping point

Chapter

12

(continued on the next page)

© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts

Reading to Learn MathematicsVocabulary Builder (continued)

NAME _____________________________ DATE _______________ PERIOD ________


Euler’s Formula

exponential series

Fibonacci sequence

fractal geometry

geometric mean

geometric sequence

geometric series

index of summation

infinite sequence

infinite series

limit

Chapter

12

(continued on the next page)

© Glencoe/McGraw-Hill ix Advanced Mathematical Concepts

Reading to Learn MathematicsVocabulary Builder (continued)

NAME _____________________________ DATE _______________ PERIOD ________


mathematical induction

n factorial

nth partial sum

orbit

Pascal’s Triangle

prisoner point

ratio test

recursive formula

sequence

sigma notation

term

trigonometric series

Chapter

12

BLANK

© Glencoe/McGraw-Hill 509 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

12-1

Arithmetic Sequences and SeriesA sequence is a function whose domain is the set of naturalnumbers. The terms of a sequence are the range elements of thefunction. The difference between successive terms of an arithmeticsequence is a constant called the common difference, denoted as d. An arithmetic series is the indicated sum of the terms of anarithmetic sequence.

Example 1 a. Find the next four terms in the arithmeticsequence �7, �5, �3, . . . .

b. Find the 38th term of this sequence.

a. Find the common difference.a2 � a1 � �5 � (�7) or 2

The common difference is 2. Add 2 to the thirdterm to get the fourth term, and so on.a4 � �3 � 2 or �1 a5 � �1 � 2 or 1a6 � 1 � 2 or 3 a7 � 3 � 2 or 5

The next four terms are �1, 1, 3, and 5.

b. Use the formula for the nth term of anarithmetic sequence.an � a1 � (n � 1)da38 � �7 � (38 �1)2 n � 38, a1 � �7, d � 2a38 � 67

Example 2 Write an arithmetic sequence that has threearithmetic means between 3.2 and 4.4.

The sequence will have the form 3.2, ? , ? , ? , 4.4.

First, find the common difference.an � a1 � (n � 1)d4.4 � 3.2 � (5 � 1)d n � 5, a5 � 4.4, a1 � 3.24.4 � 3.2 � 4dd � 0.3

Then, determine the arithmetic means.

The sequence is 3.2, 3.5, 3.8, 4.1, 4.4.

Example 3 Find the sum of the first 50 terms in the series11 � 14 � 17 � . . . � 158.

Sn � �n2�(a1 � an)

S50 � �520�(11 � 158) n � 50, a1 � 11, a50 � 158

� 4225

a2 a3 a4

3.2 � 0.3 � 3.5 3.5 � 0.3 � 3.8 3.8 � 0.3 � 4.1


Arithmetic Sequences and SeriesFind the next four terms in each arithmetic sequence.

1. �1.1, 0.6, 2.3, . . . 2. 16, 13, 10, . . . 3. p, p � 2, p � 4, . . .

For exercises 4–12, assume that each sequence or series is arithmetic.

4. Find the 24th term in the sequence for which a1 � �27 and d � 3.

5. Find n for the sequence for which an � 27, a1 � �12, and d � 3.

6. Find d for the sequence for which a1 � �12 and a23 � 32.

7. What is the first term in the sequence for which d � �3 and a6 � 5?

8. What is the first term in the sequence for which d � ��13� and a7 � �3?

9. Find the 6th term in the sequence �3 � �2�, 0, 3 � �2�, . . . .

10. Find the 45th term in the sequence �17, �11, �5, . . . .

11. Write a sequence that has three arithmetic means between 35 and 45.

12. Write a sequence that has two arithmetic means between �7 and 2.75.

13. Find the sum of the first 13 terms in the series �5 � 1 � 7 � . . . � 67.

14. Find the sum of the first 62 terms in the series �23 � 21.5 � 20 � . . . .

15. Auditorium Design Wakefield Auditorium has 26 rows, and the first row has 22seats. The number of seats in each row increases by 4 as you move toward the backof the auditorium. What is the seating capacity of this auditorium?

PracticeNAME _____________________________ DATE _______________ PERIOD ________

12-1


EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

12-1

Quadratic Formulas for SequencesAn ordinary arithmetic sequence is formed using a rule such as bn � c. The first term is c, b is called the common difference, and ntakes on the values 0, 1, 2, 3, and so on. The value of term n � 1equals b(n � 1) � c or bn � b � c. So, the value of a term is a function of the term number.

Some sequences use quadratic functions. A method called finitedifferences can be used to find the values of the terms. Notice whathappens when you subtract twice as shown in this table.

n an2 � bn � c0 c

a � b1 a � b � c

3a � b2a

2 4a � 2b � c5a � b

2a3 9a � 3b � c

7a � b2a

4 16a � 4b � cA sequence that yields a common difference after two subtractionscan be generated by a quadratic expression. For example, thesequence 1, 5, 12, 22, 35, . . . gives a common difference of 3 after twosubtractions. Using the table above, you write and solve three equa-tions to find the general rule. The equations are 1 � c,5 � a � b � c, and 12 � 4a � 2b � c.Solve each problem.

1. Refer to the sequence in the example above. Solve the system ofequations for a, b, and c and then find the quadratic expressionfor the sequence. Then write the next three terms.

2. The number of line segments connecting n points forms thesequence 0, 0, 1, 3, 6, 10, . . . , in which n is the number of pointsand the term value is the number of line segments. What is thecommon difference after the second subtraction? Find a quadraticexpression for the term value.

3. The maximum number of regions formed by n chords in a circleforms the sequence 1, 2, 4, 7, 11, 16, . . . (A chord is a line segmentjoining any two points on a circle.) Draw circles to illustrate thefirst four terms of the sequence. Then find a quadratic expressionfor the term value.


Geometric Sequences and SeriesA geometric sequence is a sequence in which each term after thefirst, a1, is the product of the preceding term and the commonratio, r. The terms between two nonconsecutive terms of ageometric sequence are called geometric means. The indicatedsum of the terms of a geometric sequence is a geometric series.Example 1 Find the 7th term of the geometric sequence

157, �47.1, 14.13, . . . .

First, find the common ratio.a2 � a1 � �47.1 � 157 or �0.3The common ratio is �0.3.

Then, use the formula for the nth term of ageometric sequence.an � a1r

n � 1

a7 � 157(�0.3)6 n � 7, a1 � 157, r � �0.3a7 � 0.114453

The 7th term is 0.114453.

Example 2 Write a sequence that has two geometric meansbetween 6 and 162.The sequence will have the form 6, ? , ? , 162.

First, find the common ratio.an � a1r

n � 1

162 � 6r3 a4 � 162, a1 � 6, n � 427 � r3 Divide each side by 6.3 � r Take the cube root of each side.

Then, determine the geometric sequence.a2 � 6 � 3 or 18 a3 � 18 � 3 or 54The sequence is 6, 18, 54, 162.

Example 3 Find the sum of the first twelve terms of thegeometric series 12 � 12�2� � 24 � 24�2� � . . . .

First, find the common ratio.a2 � a1 � � 12�2� � 12 or � �2�The common ratio is � �2�.

Sn � �a1

1�

�

ar1r

n

�

S12 ��12

1

�

�

1

(2

�

(�

�

�

2�

2�

))12

� n � 12, a1 � 12, r � ��2�

S12 � 756(1 ��2�) Simplify.

The sum of the first twelve terms of the series is 756(1 ��2�).


12-2



Geometric Sequences and Series

Determine the common ratio and find the next three terms of each geometricsequence.

1. �1, 2, �4, . . . 2. �4, �3, ��94�, . . . 3. 12, �18, 27, . . .

For exercises 4–9, assume that each sequence or series is geometric.

4. Find the fifth term of the sequence 20, 0.2, 0.002, . . . .

5. Find the ninth term of the sequence �3�, �3, 3�3�, . . . .

6. If r � 2 and a4 � 28, find the first term of the sequence.

7. Find the first three terms of the sequence for which a4 � 8.4 and r � 4.

8. Find the first three terms of the sequence for which a6 � �312� and r � �12�.

9. Write a sequence that has two geometric means between 2 and 0.25.

10. Write a sequence that has three geometric means between �32 and �2.

11. Find the sum of the first eight terms of the series �34� � �290� � �1

2070� � . . . .

12. Find the sum of the first 10 terms of the series �3 � 12 � 48 � . . . .

13. Population Growth A city of 100,000 people is growing at a rate of 5.2% per year. Assuming this growth rate remains constant, estimate the population of the city 5 years from now.

12-2



12-2

Sequences as FunctionsA geometric sequence can be defined as a function whose domain is the set of positive integers.

n � 1 2 3 4 ...↓ ↓ ↓ ↓ ↓

f (n) � ar1 – 1 ar2 – 1 ar3 – 1 ar4 – 1 ...In the exercises, you will have the opportunity to explore geometricsequences from a function and graphing point of view.

Graph each geometric sequence for n = 1, 2, 3 and 4.

1. f (n) � 2n 2. f (n) � (0.5)n

3. f (n) � (–2)n 4. f (n) � (–0.5)n

5. Describe how the graph of a geometric sequence depends on thecommon ratio.

6. Let f (n) � 2n, where n is a positive integer.a. Show graphically that for any M the graph of f (n) rises

above and stays above the horizontal line y � M.b. Show algebraically that for any M, there is a positive

integer N such that 2n � M for all n � N.



Infinite Sequences and SeriesAn infinite sequence is one that has infinitely many terms. Aninfinite series is the indicated sum of the terms of an infinitesequence.

Example 1 Find lim �4n2

n�2 �

n1� 3�.

Divide each term in the numerator and thedenominator by the highest power of n to produce anequivalent expression. In this case, n2 is the highestpower.

lim �4n2

n�2 �

n1� 3�� lim

� lim Simplify.

� Apply limit theorems.

��4 �10

��

03 � 0� or 4 lim 4 � 4, lim �n

1� � 0, lim 3 � 3,

lim �n1

2� � 0, lim 1 � 1Thus, the limit is 4.

Example 2 Find the sum of the series �32� � �38� � �332� � . . . .

In the series a1 � �32� and r � ��14�.

Since �r � < 1, S � �1a�

1

r�.

S � �1a�

1r� � a1 � �32� and r � ��4

1�

� �1120� or 1�5

1�

The sum of the series is 1�15�.

�32��1 � ��1

4��

lim 4 � lim �n1� � lim 3 � lim �n

12�

��lim 1 � lim �

n1

2�

4 � �n1� � �n

32�

��1 � �

n1

2�

�4nn2

2� � �n

n2� � �n

32�

��nn

2

2� � �n1

2�

12-3

n→∞

n→∞ n→∞ n→∞

n→∞n→∞

n→∞

n→∞

n→∞ n→∞ n→∞

n→∞ n→∞


Infinite Sequence and SeriesFind each limit, or state that the limit does not exist andexplain your reasoning.

1. limn→∞

�nn

2

2

��

11� 2. lim

n→∞�43

nn2

2��

54n�

3. limn→∞

�5n62

n� 1� 4. lim

n→∞�(n � 1

5)(n3

2n � 1)�

5. limn→∞

�3n �

4n(�

21)n

� 6. limn→∞

�n3

n�2

1�

Write each repeating decimal as a fraction.

7. 0.7�5� 8. 0.5�9�2�

Find the sum of each infinite series, or state that the sum does not exist and explain your reasoning.

9. �25� � �265� � �1

1285� � . . . 10. �34� � �18

5� � �7156� � . . .

11. Physics A tennis ball is dropped from a height of 55 feet andbounces �35� of the distance after each fall.a. Find the first seven terms of the infinite series representing

the vertical distances traveled by the ball.

b. What is the total vertical distance the ball travels before coming to rest?


12-3



12-3

Solving Equations Using SequencesYou can use sequences to solve many equations. For example,consider x2 � x – 1 � 0. You can proceed as follows.

x2 � x – 1 � 0

x(x � 1) � 1

x �

Next, define the sequence: a1 � 0 and an � .

The limit of the sequence is a solution to the original equation.

1. Let a1 � 0 and an � .

a. Write the first five terms of the sequence. Do not simplify.

b. Write decimals for the first five terms of the sequence.

c. Use a calculator to compute a6, a7, a8, and a9. Compare a9 withthe positive solution of x2 � x – 1 � 0 found by using the quadratic formula.

2. Use the method described above to find a root of 3x2 – 2x – 3 � 0.

3. Write a BASIC program using the procedure outlined above to find a root of the equation 3x2 – 2x – 3 � 0. In the program,

let a1 � 0 and an � . Run the program. Compare the

time it takes to run the program to the time it takes to evaluatethe terms of the sequence by using a calculator.

3��3an�1�2

1��1 � an � 1

1��1 � an � 1

1�1 � x


Convergent and Divergent SeriesIf an infinite series has a sum, or limit, the series is convergent. Ifa series is not convergent, it is divergent. When a series is neitherarithmetic nor geometric and all the terms are positive, you can usethe ratio test or the comparison test to determine whether theseries is convergent or divergent.

Example 1 Use the ratio test to determine whether the series �12

�12� � �22

�23� � �32

�34� � �42

�45� � . . . is convergent or divergent.

First, find the nth term. Then use the ratio test.

an � �n(n

2�n

1)� an � 1 � �

(n �21n)�

(n1� 2)

�

r � lim

r � lim �(n �21n)�

(n1� 2)

� � �n(n2�

n

1)� Multiply by the reciprocal of the divisor.

r � lim �n2�n

2� �2n2

�

n

1� � �12�

r � lim Divide by the highest power of n and apply limit theorems.

r � �12� Since r � 1, the series is convergent.

Example 2 Use the comparison test to determine whether the series

�41

2� � �

71

2� � �

1102� � �

1132� � . . . is convergent or divergent.

The general term of the series is �(3n �

11)2

�. The general

term of the convergent series 1 � �21

2� � �

31

2� � �

41

2� � . . .

is �n1

2�. Since �

(3n �

11)2

� � �n12

� for all n 1, the series

�41

2� � �

71

2� � �

1102� � �

1132� � . . . is also convergent.

1 � �n2�

�2

�(n �

21n)�

(n1� 2)

��

�n(n

2�n

1)�


12-4

Let an and an � 1 represent two consecutive terms of a series of positive terms.

Ratio Test Suppose lim �an

a�

n

1� exists and r � lim �

an

a�

n

1�. The series is convergent if r � 1 and

divergent if r � 1. If r � 1, the test provides no information.

• A series of positive terms is convergent if, for n � 1, each term of the series is equal to or less than the value of the corresponding term of some

Comparison convergent series of positive terms.Test • A series of positive terms is divergent if, for n � 1, each term of the series is

equal to or greater than the value of the corresponding term of some divergent series of positive terms.

n→∞

n→∞

n→∞

n→∞

n→∞ n→∞



Convergent and Divergent Series

Use the ratio test to determine whether each series isconvergent or divergent.

1. �12

� � �22

2

2� � �3

2

2

3� � �4

2

2

4� � . . . 2. 0.006 � 0.06 � 0.6 � . . .

3. �1 �

42 � 3� � �

1 � 28� 3 � 4� � �

1 � 2 �

136

� 4 � 5� � . . .

4. 5 � �353� � �

553��

753� � . . .

Use the comparison test to determine whether each series isconvergent or divergent.

5. 2 � �223� � �

323� � �

423� � . . . 6. �52� � 1 � �58� � �1

51� � . . .

7. Ecology A landfill is leaking a toxic chemical. Six months afterthe leak was detected, the chemical had spread 1250 meters fromthe landfill. After one year, the chemical had spread 500 metersmore, and by the end of 18 months, it had reached an additional200 meters.a. If this pattern continues, how far will the chemical spread

from the landfill after 3 years?

b. Will the chemical ever reach the grounds of a hospital located2500 meters away from the landfill? Explain.

12-4



12-4

Alternating SeriesThe series below is called an alternating series.

1 – 1 � 1 – 1 � ...

The reason is that the signs of the terms alternate. An interestingquestion is whether the series converges. In the exercises, you willhave an opportunity to explore this series and others like it.

1. Consider 1 – 1 � 1 – 1 � ... .a. Write an argument that suggests that the sum is 1.

b. Write an argument that suggests that the sum is 0.

c. Write an argument that suggests that there is no sum.(Hint: Consider the sequence of partial sums.)

If the series formed by taking the absolute values of the terms of agiven series is convergent, then the given series is said to beabsolutely convergent. It can be shown that any absolutely convergent series is convergent.

2. Make up an alternating series, other than a geometric series withnegative common ratio, that has a sum. Justify your answer.



12-5

Sigma Notation and the nth TermA series may be written using sigma notation.

maximum value of n →�an ← expression for general term

starting value of n →

index of summation

Example 1 Write each expression in expanded formand then find the sum.

a. � (n � 2)

First, write the expression in expanded form.

� (n � 2) � (1 � 2) � (2 � 2) � (3 � 2) � (4 � 2) � (5 � 2)

Then, find the sum by simplifying the expandedform. 3 � 4 � 5 � 6 � 7 � 25

b. � 2��14��m

� 2��14��m

� 2��14��1� 2��14��2

� 2��14��3� . . .

� �12� � �18� � �312� � . . .

This is an infinite series. Use the formula S � �1a�

1

r�.

S � a1 � �12�, r � �14�

S � �32�

Example 2 Express the series 26 � 37 � 50 � 65 � . . . � 170using sigma notation.

Notice that each term is one more than a perfectsquare. Thus, the nth term of the series is n2 � 1.Since 52 � 1 � 26 and 132 � 1 � 170, the index ofsummation goes from n � 5 to n � 13.

Therefore, 26 � 37 � 50 � 65 �. . . � 170 � � (n2 � 1).

�12

�

�1 � �14�

k

n�1

5

n�1

5

n�1

∞

m�1∞

m�1

13

n�5

↑


Sigma Notation and the nth TermWrite each expression in expanded form and then find the sum.

1. �5

n�3(n2 � 2n) 2. �

4

q�1�q2�

3. �5

t�1t(t � 1) 4. �

3

t�0(2t � 3)

5. �5

c�2(c � 2)2 6. �

∞

i�110��12��

i

Express each series using sigma notation.

7. 3 � 6 � 9 � 12 � 15 8. 6 � 24 � 120 � . . . � 40,320

9. �11� � �14� � �19� � . . . � �1010� 10. 24 � 19 � 14 � . . . � (�1)

11. Savings Kathryn started saving quarters in a jar. She beganby putting two quarters in the jar the first day and then sheincreased the number of quarters she put in the jar by one additional quarter each successive day.a. Use sigma notation to represent the total number of quarters

Kathryn had after 30 days.

b. Find the sum represented in part a.


12-5


NAME _____________________________ DATE _______________ PERIOD ________

Street Networks: Finding All Possible RoutesA section of a city is laid out in square blocks. Going north from the intersection of 1st Avenue and 1st Street, the avenues are 1st, 2nd, 3rd, and so on. Going east, the streets are numbered in the same way.

Factorials can be used to find the number,r(e, n), of different routes between two intersections.

The number of streets going east is e; the number of avenues goingnorth is n.

The following problems examine the possible routes from one location to another. Assume that you never use a route that is unnecessarily long. Assume that e 1 and n 1.

Solve each problem.

1. List all the possible routes from 1st Street and 1st Avenue to 4th Street and 3rd Avenue. Use ordered pairs to show the routes,with street numbers first and avenue numbers second. Each route must start at (1, 1) and end at (4, 3).

2. Use the formula to compute the number of routes from (1, 1) to (4, 3).There are 4 streets going east and 3 avenues going north.

3. Find the number of routes from 1st Street and 1st Avenue to 7th Street and 6th Avenue.

Enrichment12-5

r(e, n) �[(e � 1) � (n � 1)]!��

(e � 1)! (n � 1)!


The Binomial TheoremTwo ways to expand a binomial are to use either Pascal’striangle or the Binomial Theorem. The Binomial Theoremstates that if n is a positive integer, then the following is true.

(x � y)n � xn � nxn � 1y � �n(

1n

�

�

21)

� xn � 2y2 ��n(n

1�

�

12)(

�

n3� 2)

� xn � 3y3 � . . . � yn

To find individual terms of an expansion, use this form of theBinomial Theorem:

(x � y)n � � �r!(nn�!

r)!� xn � ryr.

Example 1 Use Pascal’s triangle to expand (x � 2y)5.

First, write the series without the coefficients. Theexpression should have 5 � 1, or 6, terms, with the firstterm being x5 and the last term being y5. The exponents of xshould decrease from 5 to 0 while the exponents of y shouldincrease from 0 to 5. The sum of the exponents of each termshould be 5.

x5 � x4y � x3y2 � x2y3 � xy4 � y5 x0 � 1 and y0 � 1

Replace each y with 2y.

x5 � x4(2y) � x3(2y)2 � x2(2y)3 � x(2y)4 � (2y)5

Then, use the numbers in the sixth row of Pascal’s triangleas the coefficients of the terms, and simplify each term.

1 5 10 10 5 1↓ ↓ ↓ ↓ ↓ ↓

(x � 2y)5 � x5 � 5x4(2y) � 10x3(2y)2 � 10x2(2y)3 � 5x(2y)4 � (2y)5

� x5 � 10x4y � 40x3y2 � 80x2y3 � 80xy4 � 32y5

Example 2 Find the fourth term of (5a � 2b)6.

(5a � 2b)6 � � �r!(66�!

r)!� (5a)6 � r(2b)r

To find the fourth term, evaluate the general term for r � 3.Since r increases from 0 to n, r is one less than the numberof the term.

�r!(66�!

r)!�(5a)6 � r(2b)r � �3!(66�!

3)!� (5a)6 � 3(2b)3

� �6 � 53

�!3

4!

� 3!� (5a)3(2b)3

� 20,000a3b3

The fourth term of (5a � 2b)6 is 20,000a3b3.


12-6

n

r�0

6

r�0



The Binomial Theorem

Use Pascal’s triangle to expand each binomial.

1. (r � 3)5 2. (3a � b)4

Use the Binomial Theorem to expand each binomial.3. (x � 5)4 4. (3x � 2y)4

5. (a � �2�)5 6. (2p � 3q)6

Find the designated term of each binomial expansion.7. 4th term of (2n � 3m)4 8. 5th term of (4a � 2b)8

9. 6th term of (3p � q)9 10. 3rd term of (a � 2�3�)6

11. A varsity volleyball team needs nine members. Of these ninemembers, at least f ive must be seniors. How many of the possible groups of juniors and seniors have at least f ive seniors?

12-6


Patterns in Pascal’s TriangleYou have learned that the coefficients in the expansion of (x � y)n

yield a number pyramid called Pascal’s triangle.

Row 1

Row 2

Row 3

Row 4

Row 5

Row 6

Row 7

As many rows can be added to the bottom of the pyramid as you need.This activity explores some of the interesting properties of this famous number pyramid.1. Pick a row of Pascal’s triangle.

a. What is the sum of all the numbers in all the rows above the row you picked?

b. What is the sum of all the numbers in the row you picked?

c. How are your answers for parts a and b related?

d. Repeat parts a through c for at least three more rows of Pascal’s triangle. What generalization seems to be true?

e. See if you can prove your generalization.

2. Pick any row of Pascal’s triangle that comes after the first.a. Starting at the left end of the row, find the sum of the odd

numbered terms.

b. In the same row, find the sum of the even numbered terms.

c. How do the sums in parts a and b compare?

d. Repeat parts a through c for at least three other rows ofPascal’s triangle. What generalization seems to be true?


12-6

→

→

→

→

→

→

→

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1



Special Sequences and SeriesThe value of ex can be approximated by using theexponential series. The trigonometric series can be usedto approximate values of the trigonometric functions. Euler’sformula can be used to write the exponential form of acomplex number and to find a complex number that is thenatural logarithm of a negative number.

Example 1 Use the first five terms of the trigonometricseries to approximate the value of sin ��6� tofour decimal places.

sin x � x � �x33

!� � �x55

!� � �x77

!� � �9x9

!�

Let x � ��6�, or about 0.5236.

sin ��6� � 0.5236 � �(0.5

32!36)3� � �

(0.552!36)5� � �

(0.572!36)7� � �

(0.592!36)9�

sin ��6� � 0.5236 � 0.02392 � 0.00033 � 0.000002 � 0.000000008

sin ��6� � 0.5000 Compare this result to the actual value, 0.5.

Example 2 Write 4 � 4i in exponential form.

Write the polar form of 4 � 4i.Recall that a � bi � r(cos � � i sin �), where r � �a�2�� b�2� and � � Arctan �a

b� when a � 0.

r � �4�2�� (��4�)2� or 4�2�, and a � 4 and b � �4

� � Arctan ��44� or ��4�

4 � 4i � 4�2��cos ��4�� i sin ��4�� 4�2�e

�i��4�

Thus, the exponential form of 4 � 4i is 4�2�e�i�

�4�.

Example 3 Evaluate ln(�12.4).

ln(�12.4) � ln(�1) � ln(12.4)� i� � 2.5177 Use a calculator to compute ln(12.4).

Thus, ln(�12.4) � i� � 2.5177. The logarithm is a complex number.

12-7


Special Sequences and Series

Find each value to four decimal places.

1. ln(�5) 2. ln(�5.7) 3. ln(�1000)

Use the first five terms of the exponential series and a calculatorto approximate each value to the nearest hundredth.

4. e0.5 5. e1.2

6. e2.7 7. e0.9

Use the first five terms of the trigonometric series to approximatethe value of each function to four decimal places. Then, comparethe approximation to the actual value.

8. sin �56�� 9. cos �34

��

Write each complex number in exponential form.

10. 13�cos ��3� � i sin ��3�� 11. 5 � 5i

12. 1 � �3�i 13. �7 � 7�3�i

14. Savings Derika deposited $500 in a savings account with a4.5% interest rate compounded continuously. (Hint: The formulafor continuously compounded interest is A � Pert.)a. Approximate Derika’s savings account balance after 12 years

using the first four terms of the exponential series.

b. How long will it take for Derika’s deposit to double, providedshe does not deposit any additional funds into her account?


12-7



Power SeriesA power series is a series of the form

a0 � a1x � a2x2 � a3x3 � ...

where each ai is a real number. Many functions can be represented by power series. For instance, the function f (x) � ex can be represented by the series

ex � 1 � x � � � ... .

Use a graphing calculator or computer to graph the functions in Exercies 1–4.

1. f2(x) � 1 � x 2. f3(x) � 1 � x �

3. f4(x) � 1 � x � � 4. f5(x) � 1 � x � � �

5. Write a statement that relates the sequence of graphs suggested by Exercies 1–4 and the function y � ex.

6. The series 1 � x2 � x4 � x6 � ... is a power series for which each ai � 1. The series is also a geometric series with first term 1 andcommon ratio x2.a. Find the function that this power series represents.

b. For what values of x does the series give the values of the function in part a?

7. Find a power series representation for the function f (x) � .3

�1 � x2

x4

�4!

x3

�3!

x2

�2!

x3

�3!

x2

�2!

x2

�2!

x3

�3!

x2

�2!

12-7


12-8

Sequences and IterationEach output of composing a function with itself is called aniterate. To iterate a function ƒ(x), find the function value ƒ(x0)of the initial value x0. The second iterate is the value of thefunction performed on the output, and so on.

The function ƒ(z) � z2 � c, where c and z are complex numbers,is central to the study of fractal geometry. This type ofgeometry can be used to describe things such as coastlines,clouds, and mountain ranges.

Example 1 Find the first four iterates of the functionƒ(x) � 4x � 1 if the initial value is �1.

x0 � �1x1 � 4(�1) � 1 or �3x2 � 4(�3) � 1 or �11x3 � 4(�11) � 1 or �43x4 � 4(�43) � 1 or �171

The first four iterates are �3, �11, �43, and �171.

Example 2 Find the first three iterates of the functionƒ(z) � 3z � i if the initial value is 1 � 2i.

z0 � 1 � 2iz1 � 3(1 � 2i) � i or 3 � 5iz2 � 3(3 � 5i) � i or 9 � 14iz3 � 3(9 � 14i) � i or 27 � 41i

The first three iterates are 3 � 5i, 9 � 14i, and 27 � 41i.

Example 3 Find the first three iterates of the functionƒ(z) � z2 � c, where c � 2 � i and z0 � 1 � i.

z1 � (1 � i)2 � 2 � i.� 1 � i � i � i2 � 2 � i� 1 � i � i � (�1) � 2 � i i2 � �1� 2 � i

z2 � (2 � i)2 � 2 � i� 4 � 2i � 2i � i2 � 2 � i� 4 � 2i � 2i � (�1) � 2 � i� 5 � 3i

z3 � (5 � 3i)2 � 2 � i� 25 � 15i � 15i � 9i2 � 2 � i� 25 � 15i � 15i � 9(�1) � 2 � i� 18 � 29i

The first three iterates are 2 � i, 5 � 3i, and 18 � 29i.




Sequences and Iteration

Find the first four iterates of each function using the given initial value. If necessary, round your answers to the nearest hundredth.

1. ƒ(x) � x2 � 4; x0 � 1 2. ƒ(x) � 3x � 5; x0 � �1

3. ƒ(x) � x2 � 2; x0 � �2 4. ƒ(x) � x(2.5 � x); x0 � 3

Find the first three iterates of the function ƒ(z) � 2z � (3 � i) for each initial value.

5. z0 � i 6. z0 � 3 � i

7. z0 � 0.5 � i 8. z0 � �2 � 5i

Find the first three iterates of the function ƒ(z) � z2 � c for eachgiven value of c and each initial value.

9. c � 1 � 2i; z0 � 0 10. c � i; z0 � i

11. c � 1 � i; z0 � �1 12. c � 2 � 3i; z0 � 1 � i

13. Banking Mai deposited $1000 in a savings account. Theannual yield on the account is 5.2%. Find the balance of Mai’saccount after each of the f irst 3 years.

12-8


DepreciationTo run a business, a company purchases assets such as equipment orbuildings. For tax purposes, the company distributes the cost of these assets as a business expense over the course of a number ofyears. Since assets depreciate (lose some of their market value) as they get older, companies must be able to figure the depreciationexpense they are allowed to take when they file their income taxes.

Depreciation expense is a function of these three values:1. asset cost, or the amount the company paid for the asset;2. estimated useful life, or the number of years the company can

expect to use the asset;3. residual or trade-in value, or the expected cash value of the

asset at the end of its useful life.

In any given year, the book value of an asset is equal to the asset cost minus the accumulated depreciation. This value represents theunused amount of asset cost that the company may depreciate infuture years. The useful life of the asset is over once its book valueis equal to its residual value.

There are several methods of determining the amount of depreciation in a given year. In the declining-balance method, thedepreciation expense allowed each year is equal to the book value ofthe asset at the beginning of the year times the depreciation rate.Since the depreciation expense for any year is dependent upon thedepreciation expense for the previous year, the process of determining the depreciation expense for a year is an iteration.

The table below shows the first two iterates of the depreciation schedule for a $2500 computer with a residual value of $500 if the depreciation rate is 40%.

1. Find the next two iterates for the depreciation expense function.

2. Find the next two iterates for the end-of-year book value function.

3. Explain the depreciation expense for year 5.


12-8

End of Asset Depreciation Book Value atYear Cost Expense End of Year

1 $2500 $1000 $1500(40% of $2500) ($2500 - $1000)

2 $2500 $600 $900(40% of $1500) ($1500 - $600)



Mathematical InductionA method of proof called mathematical induction can beused to prove certain conjectures and formulas. The followingexample demonstrates the steps used in proving a summationformula by mathematical induction.

Example Prove that the sum of the first n positiveeven integers is n(n � 1).

Here Sn is defined as 2 � 4 � 6 � . . . � 2n � n(n � 1).

1. First, verify that Sn is valid for the first possible case, n � 1.Since the first positive even integer is 2 and 1(1 � 1) � 2, the formula is valid for n � 1.

2. Then, assume that Sn is valid for n � k.

Sk ⇒ 2 � 4 � 6 � . . . � 2k � k(k � 1). Replace n with k.

Next, prove that Sn is also valid for n � k � 1.

Sk � 1 ⇒ 2 � 4 � 6 � . . . � 2k � 2(k � 1)

� k(k � 1) � 2(k � 1) Add 2(k � 1) to both sides.

We can simplify the right side by adding k(k � 1) � 2(k � 1).

Sk � 1 ⇒ 2 � 4 � 6 � . . . � 2k � 2(k � 1)

� (k � 1)(k � 2) (k � 1) is a common factor.

If k � 1 is substituted into the original formula (n(n � 1)),the same result is obtained.

(k � 1)[(k � 1) � 1] or (k � 1)(k � 2)

Thus, if the formula is valid for n � k, it is also validfor n � k � 1. Since Sn is valid for n � 1, it is alsovalid for n � 2, n � 3, and so on. That is, the formulafor the sum of the first n positive even integers holds.

12-9


Mathematical InductionUse mathematical induction to prove that each proposition is validfor all positive integral values of n.

1. �13� � �23� � �33� � . . . � �n3� � �

n(n6� 1)�

2. 5n � 3 is divisible by 4.


12-9



Conjectures and Mathematical InductionFrequently, the pattern in a set of numbers is not immediately evident. Once you make a conjecture about a pattern, you can usemathematical induction to prove your conjecture.

1. a. Graph f (x) � x2 and g(x) � 2x on the axes shown at the right.

b. Write a conjecture that compares n2 and 2n, where n is a positive integer.

c. Use mathematical induction to prove your response from part b.

2. Refer to the diagrams at the right.a. How many dots would there be in the fourth

diagram S4 in the sequence?

b. Describe a method that you can use to determine the number of dots in the fifth diagram S5 based on the number of dots in the fourth diagram, S4. Verify your answer by constructing the fifth diagram.

c. Find a formula that can be used to compute the number of dots in the nth diagram of this sequence.Use mathematical induction to prove your formula iscorrect.

12-9

S1 S2 S3

BLANK


Chapter 12 Test, Form 1A

NAME _____________________________ DATE _______________ PERIOD ________Chapter

12Write the letter for the correct answer in the blank at the right of each problem.

1. Find the 15th term in the arithmetic sequence 14, 10.5, 7, . . . . 1. ________A. �21 B. �63 C. 63 D. �35

2. Find the sum of the first 36 terms in the arithmetic series 2. ________�0.2 � 0.3 � 0.8 � . . . .A. 318.6 B. 332.2 C. 307.8 D. 315

3. Find the sixth term in the geometric sequence �3� y3, �3y5, 3�3� y7, . . . . 3. ________A. �27y13 B. 9�3� y13 C. 27y13 D. �9�3� y13

4. Find the sum of the first f ive terms in the geometric series 4. ________��32� � 1 � �23� � . . . .

A. �5554� B. ��55

54� C. �52

57� D. ��52

57�

5. Find three geometric means between ��2� and �4�2�. 5. ________A. 2, �2�2�, 4 B. �2, 2�2�, �4C. 2, 2�2�, 4 D. A or C

6. Find limn→∞ �1 � �

(�n1)n

� . 6. ________

A. 1 B. 0 C. �1 D. does not exist

7. Find the sum of �2�7� � �9� � �3� � . . . . 7. ________

A. �12�(9 � 9 �3�) B. 9 � 9�3� C. �21�(9 � 9 �3�) D. does not exist

8. Write 3.1�2�3� as a fraction. 8. ________

A. �13034303� B. �13

03430� C. �10

3430� D. �1

3034303�

9. Which of the following series is convergent? 9. ________A. �3� � 3 � 3�3� � . . . B. 6�2� � 12 � 12�2� � . . .C. 6�2� � 6 � 3�2� � . . . D. 6�2� � 12 � 12�2� � . . .

10. Which of the following series is divergent? 10. ________

A. 1 � 3��14�� 9��14��2

� 27��14��3

� . . . B. 1 � 3��15�� 9��15��2

� 27��15��3

� . . .

C. 1 � 3��17�� 9��17��2

� 27��17��3

� . . . D. 1 � 3��12�� 9��12��2

� 27��12��3

� . . .

11. Write �3

k�0��2

1��k

in expanded form and then find the sum. 11. ________

A. ��12� � �14� � �18�; ��78� B. 1 � �12� � �14� � �18�; �18�

C. ��12� � �14� � �18�; �83� D. 1 � �12� � �14� � �18�; �58�


12. Express the series �27 � 9 � 3 � 1 � . . . using sigma notation. 12. ________

A. �∞

k�0�3k B. �

3

k�0�27��13��

k

C. �∞

k�0�27��13��

kD. �

∞

k�027��13��

k

13. The expression 81p4 � 108p3r3 � 54p2r6 � 12pr9 � r12 is the 13. ________expansion of which binomial?A. ( p � 3r3)4 B. (3p � r3)4 C. (3p3 � r)4 D. (3p � 3r3)4

14. Find the fifth term in the expansion of (3x2 � �y�)6. 14. ________A. 135x4y2 B. 45x4y2 C. �135x4y2 D. �45x4y2

15. Use the first f ive terms of the trigonometric series to find the value 15. ________of sin �1

�2� to four decimal places.

A. 0.2618 B. 0.2588 C. 0.7071 D. 0.2648

16. Find ln (�91.48). 16. ________A. 4.5161 B. i� � 4.5161 C. i� � 4.5161 D. �4.5161

17. Write 3 � �3�i in exponential form. 17. ________A. 9ei�

116�� B. 9ei�

53�� C. 2�3�ei�

116�� D. 2�3�ei�

53��

18. Find the first three iterates of the function ƒ(z) � �z � i for 18. ________z0 � 2 � 3i.A. 2 � 2i, 2 � 3i, 2 � 2i B. �2 � 2i, 2 � 3i, �2 � 2iC. 2 � 3i, �2 � 2i, 2 � 3i D. �2 � 2i, 2 � 3i, �2 � 2i

19. Find the first three iterates of the function ƒ(z) � z2 � c for 19. ________c � 1 � 2i and z0 � 1 � i.A. �1 � 4i, �15 � 8i, 219 � 194iB. �1 � 4i, �16 � 6i, 220 � 192iC. �1 � 4i, �16 � 6i, 221 � 194iD. �1 � 4i, �16 � 6i, 219 � 194i

20. Suppose in a proof of the summation formula 20. ________1 � 5 � 25 � . . . � 5n�1 � �14�(5n � 1) by mathematical induction,you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k � 1) � �14� (5k�1 � 1)

B. 1 � 5 � 25 � . . . � 5k � 5k�1 � �14� (5k � 1) � 5k�1�1

C. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k�1 � 1) � 5k�1�1

D. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k � 1) � 5k�1�1

Bonus Solve �6

n�0(3n � 2x) � 7 for x. Bonus: ________

A. �121� B. 8 C. 4 D. �13

9�

Chapter 12 Test, Form 1A (continued)


12


Chapter 12 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. Find the 27th term in the arithmetic sequence �8, 1, 10, . . . . 1. ________A. 174 B. 242 C. 235 D. 226

2. Find the sum of the first 20 terms in the arithmetic series 2. ________14 � 3 � 8 � . . . .A. �195 B. �1810 C. 195 D. 1810

3. Find the sixth term in the geometric sequence 11, �44, 176, . . . . 3. ________A. 11,264 B. �11,264 C. 45,056 D. �45,056

4. Find the sum of the first five terms in the geometric series 4. ________2 � �43� � �89� � . . . .

A. �8515� B. �12

37� C. �18

110� D. �28

715�

5. Find three geometric means between ��23� and �54. 5. ________A. 2, 6, 18 B. �2, 6, �18 C. 2, �6, 18 D. A or C

6. Find lim �5n43n�

3 �7n

72n�

2

3�. 6. ________

A. �54� B. 0 C. �45� D. does not exist

7. Find the sum of �151� � �35

35� � �6

9095� � . . . . 7. ________

A. �17201� B. ��17

201� C. �27

2� D. does not exist

8. Write 0.1�2�3� as a fraction. 8. ________A. �43

13� B. �3

43133� C. �3

4313� D. �34

313�

9. Which of the following series is convergent? 9. ________A. 7.5 � 1.5 � 0.3 � . . . B. 1.2 � 3.6 � 10.8 � . . .C. 1.2 � 3.6 � 10.8 � . . . D. �2.5 � 2.5 � 2.5 � . . .

10. Which of the following series is divergent? 10. ________A. �3

12� � �6

12� � �9

12� � . . . B. �23

2� � �26

4� � �29

6� � . . .

C. �13�12� � �23

�23� � �33

�34� � . . . D. �0.

352� � �0.

654� � �0.

956� � . . .

11. Write �5��23��kin expanded form and then find the sum. 11. ________

A. 5��23��2� ��23��2

� ��23��2; �29

8� B. ��5 3� 2��2

� ��5 3� 2��3

� ��5 3� 2��4

; �15

8,7100

�

C. 5��23��1� 5��23��2

� 5��23��3; �12

970� D. 5��23��2

� 5��23��3� 5��23��4

; �38810�

Chapter

12

k�2

4


12. Express the series 0.7 � 0.007 � 0.00007 � . . . using sigma notation. 12. ________

A. �0.7(10)k � 1 B. �7(10)1 � 2k C. �7(10)1 � k D. �0.7(10)�k

13. The expression 243c5 � 810c4d � 1080c3d2 � 720c2d3 � 240cd4 � 32d5 13. ________is the expansion of which binomial?A. (3c � d)5 B. (c � 2d)5 C. (2c � 3d)5 D. (3c � 2d)5

14. Find the third term in the expansion of (3x � y)6. 14. ________A. 1215x4y2 B. 1215x2y4 C. �1215x2y4 D. �1215x4y2

15. Use the first five terms of the exponential series 15. ________ex � 1 � x � �x2

2

!� ��x33

!� � �x44

!� � . . . to approximate e3.9.

A. 39.40 B. 24.01 C. 32.03 D. 90.11

16. Find ln (�102). 16. ________A. 4.6250 B. i� � 4.6250 C. i� � 4.6250 D. �4.6250

17. Write 15�3� � 15i in exponential form. 17. ________

A. 30ei�11

6�� B. 30ei�

56�� C. 30ei�

76�� D. 15ei�

116��

18. Find the first three iterates of the function ƒ(z) � �2z for z0 � 1 � 3i. 18. ________A. 2 � 6i, 4 � 12i, 8 � 24i B. �2 � 6i, 4 � 12i, �8 � 24iC. �2 � 6i, 4 � 12i, 8 � 24i D. 2 � 6i, �4 � 12i, 8 � 24i

19. Find the first three iterates of the function ƒ(z) � z2 � c for c � i 19. ________and z0 � 1.A. 1 � i, 2 � 3i, �5 � 13i B. 1 � i, �3i, �9 � iC. 1 � i, �3i, 9 � i D. 1 � i, �2i, �4 � i

20. Suppose in a proof of the summation formula 7 � 9 � 11 � . . . � 20. ________2n � 5 � n(n � 6) by mathematical induction, you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 7 � 9 � 11 � . . . � 2k � 5 � 2(k � 1) � 5 � k(k � 6) � (k � 1)(k � 1 � 6)B. 7 � 9 � 11 � . . . � 2(k � 1) � 5 � k(k � 6)C. 7 � 9 � 11 � . . . � 2k � 5 � k(k � 6)D. 7 � 9 � 11 � . . . � 2k � 5 � 2(k � 1) � 5 � k(k � 6) � 2(k � 1) � 5

Bonus If a1, a2, a3, . . ., an is an arithmetic sequence, where an 0, Bonus: ________then �a

11�, �a

12�, �a

13�, . . ., �a

1n� is a harmonic sequence. Find one

harmonic mean between �12� and �18�.

A. �14� B. �15� C. �16� D. �156�

Chapter 12 Test, Form 1B (continued)


12

∞

k�1

∞

k�1

∞

k�1

∞

k�1


Chapter 12 Test, Form 1C

NAME _____________________________ DATE _______________ PERIOD ________


1. Find the 21st term in the arithmetic sequence 9, 3, �3, . . . . 1. ________A. �111 B. �129 C. �117 D. �126

2. Find the sum of the first 20 terms in the arithmetic series 2. ________�6 � 12 � 18 � . . . .A. �2520 B. �1266 C. �1140 D. �1260

3. Find the 10th term in the geometric sequence �2, 6, �18, . . . . 3. ________A. 118,098 B. �118,098 C. 39,366 D. �39,366

4. Find the sum of the first eight terms in the geometric series 4. ________�4 � 8 � 16 � . . . .A. �342 B. �1020 C. �340 D. 340

5. Find one geometric mean between 2 and 32. 5. ________A. �16 B. 8 C. 12 D. 4

6. Find limn→∞

�n3

n�2

5� . 6. ________

A. �23� B. �5 C. �85� D. does not exist

7. Find the sum of 16 � 4 � 1 � . . . . 7. ________

A. 64 B. �654� C. 20 D. does not exist

8. Write 0.8� as a fraction. 8. ________

A. �98989� B. �9

8� C. �989� D. �8

9�

9. Which of the following series is convergent? 9. ________A. 8 � 8.8 � 9.68 � . . . B. 8 � 6 � 4 � . . .C. 8 � 2.4 � 0.72 � . . . D. 8 � 8 � 8 � . . .

10. Which of the following series is divergent? 10. ________

A. 1 � �21

2� � �

21

4� � �

21

6� � . . . B. 1 � �

31

2� � �

31

4� � �

31

6� � . . .

C. 1 � �21

2� � �

21

4� � �

21

6� � . . . D. 1 � ��32��

2� ��32��

4� ��32��

6� . . .

11. Write �4

k�13k�1 in expanded form and then find the sum. 11. ________

A. 1 � 3 � 9 � 27; 40 B. 1 � �13� � �19� � �217�; �42

07�

C. 3 � 9 � 27 � 81; 120 D. 0 � 2 � 8 � 26; 36

Chapter

12


12. Express the series 5 � 9 � 13 � . . . � 101 using sigma notation. 12. ________

A. �∞

k�1(4k � 1) B. �

25

k�1(4k � 1) C. �

25

k�1(4k � 1) D. �

24

k�1(4k � 1)

13. The expression 32x5 � 80x4 � 80x3 � 40x2 � 10x � 1 is the 13. ________expansion of which binomial?A. (2x � 1)5 B. (x � 2)5 C. (2x � 2)5 D. (2x � 1)5

14. Find the fourth term in the expansion of (3x � y)7. 14. ________A. 105x4y3 B. 420x4y3 C. 1701x4y3 D. 2835x4y3

15. Use the first f ive terms of the exponential series 15. ________ex � 1 � x � �x2

2

!� � �x33

!� � �x44

!� � . . . to approximate e5.

A. 65.375 B. 148.41 C. 48.41 D. 76.25

16. Find ln (�21). 16. ________A. 3.0445 B. i� � 3.0445 C. i� � 3.0445 D. �3.0445

17. Write 1 � i in exponential form. 17. ________A. �2� e i�

�4� B. �2�ei�

74�� C. ei�

74�� D. ei�

�4�

18. Find the first three iterates of the function ƒ(z) � z � i for z0 � 1. 18. ________A. 1, 1 � i, 1 � 2i B. 1 � i, 2 � 2i, 3 � 3iC. 1 � i, 1 � 2i, 1 � 3i D. 1 � i, 1 � i, 1 � i

19. Find the first three iterates of the function ƒ(z) � z2 � c for c � i 19. ________and z0 � i.A. �1 � i, i, �1�i B. �1 � i, 3i, �9C. 1 � i, �3i, 9 � i D. �1 � i, 2i, �4 � i

20. Suppose in a proof of the summation formula 20. ________1 � 5 � 9 � . . . � 4n � 3 � n(2n � 1) by mathematical induction,you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 1 � 5 � 9 � . . . � 4k � 3 � 4(k � 1) � 3 � k(2k � 1) � 4(k � 1) � 3B. 1 � 5 � 9 � . . . � 4k � 3 � k(2k � 1) � 4(k � 1) � 3C. 1 � 5 � 9 � . . . � 4k � 3 � k(2k � 1)D. 1 � 5 � 9 � . . . � 4k � 3 � 4(k � 1) � 3 � k(2k � 1) � (k � 1)[2(k � 1) � 1]

Bonus If a1, a2, a3, . . . , an is an arithmetic sequence, where an 0, Bonus: ________

then �a1

1�, �a

12�, �a

13�, . . . , �a

1n� is a harmonic sequence. Find one

harmonic mean between 2 and 3.

A. �25� B. �52� C. �152� D. �15

2�

Chapter 12 Test, Form 1C (continued)


12


Chapter 12 Test, Form 2A

NAME _____________________________ DATE _______________ PERIOD ________

1. Find d for the arithmetic sequence in which a1 � 14 and 1. __________________a28 � 32.

2. Find the 15th term in the arithmetic sequence 2. __________________11�45�, 10 �25�, 9, 7�35�, . . . .

3. Find the sum of the first 27 terms in the arithmetic series 3. __________________35.5 � 34.3 � 33.1 � 31.9 � . . . .

4. Find the ninth term in the geometric sequence 25, 10, 4, . . . . 4. __________________

5. Find the sum of the first eight terms in the geometric series 5. __________________�15� � 2 � 20 � . . . .

6. Form a sequence that has three geometric means between 6. __________________6 and 54.

7. Find limn→∞

�9n

133n

�

4

5�

n25n

�

2

4� or state that the limit does not exist. 7. __________________

8. Find the sum of the series 6�2� � 6 � 3�2� � 3 � . . . or 8. __________________state that the sum does not exist.

9. Write 0.06�4� as a fraction. 9. __________________

Determine whether each series is convergent or divergent.

10. �2�2�

1� 13� � �

2�2�1

� 23� � �2�2�

1� 33� � . . . 10. __________________

11. �211� � �22

2� � �23

3� � . . . 11. __________________

Chapter

12


12. Write �7

k�227��13��

k�2in expanded form and then find 12. __________________

the sum.

13. Express the series �31�09� � �31

�211� � �31

�413� � . . . � �32

�423� 13. __________________

using sigma notation.

14. Use the Binomial Theorem to expand (1 � �3�)5. 14. __________________

15. Find the fifth term in the expansion of (3x3 � 2y2)5. 15. __________________

16. Use the first f ive terms of the exponential series 16. __________________to approximate e2.7.

17. Find ln (�12.7) to four decimal places. 17. __________________

18. Find the first three iterates of the function ƒ(z) � 3z � 1 18. __________________for z0 � 2 � i.

19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � 1 � i and z0 � 2i.

20. Use mathematical induction to prove that 20. __________________1 � 5 � 25 � . . . � 5n�1 � �4

1� (5n � 1). Write your proof on a separate piece of paper.

Bonus If ƒ(z) � z2 � z � c is iterated with an initial Bonus: __________________value of 3 � 4i and z1 � 4 � 11i, f ind c.

Chapter 12 Test, Form 2A (continued)


12


Chapter 12 Test, Form 2B

NAME _____________________________ DATE _______________ PERIOD ________

1. Find d for the arithmetic sequence in which a1 � 6 1. __________________and a13 � �42.

2. Find the 40th term in the arithmetic sequence 2. __________________7, 4.4, 1.8, �0.8, . . . .

3. Find the sum of the first 30 terms in the arithmetic series 3. __________________10 � 6 � 2 � 2 � . . . .

4. Find the ninth term in the geometric sequence �217�, �19�, �13�, . . . . 4. __________________

5. Find the sum of the first eight terms in the geometric series 5. __________________64 � 32 � 16 � 8 � . . . .

6. Form a sequence that has three geometric means between 6. __________________�4 and �324.

7. Find lim �2nn�3


8. Find the sum of the series 12 � 8 � �136� � . . . or state 8. __________________

that the sum does not exist.



10. �41� � �17� 2� � �1 �

120

� 3� � . . . 10. __________________

11. �311� � �32

2� � �33

3� � . . . 11. __________________

Chapter

12


12. Write �(k � 1)(k � 2) in expanded form and then find the 12. __________________sum.

13. Express the series �1 2� 0� � �2 3

� 1� � �3 4� 2� � . . . � �10

11� 9� using 13. __________________

sigma notation.

14. Use the Binomial Theorem to expand (2p � 3q)4. 14. __________________

15. Find the fifth term in the expansion of (4x � 2y)7. 15. __________________

16. Use the first five terms of the cosine series 16. __________________cos x � 1 � �x2

2

!� � �x44

!� � �x66

!� � �x88

!� � . . . to approximate

the value of cos �4�� to four decimal places.

17. Find ln (�13.4) to four decimal places. 17. __________________

18. Find the first three iterates of the function ƒ(z) � 0.5z 18. __________________for z0 � 4 � 2i.

19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � 2i and z0 � 1.

20. Use mathematical induction to prove that 7 � 9 � 11 � . . . � 20. __________________(2n � 5) � n(n � 6). Write your proof on a separate piece of paper.

Bonus Find the sum of the coefficients of the expansion Bonus: __________________of (x � y)7.

Chapter 12 Test, Form 2B (continued)


123

k�0


Chapter 12 Test, Form 2C

NAME _____________________________ DATE _______________ PERIOD ________

1. Find d for the arithmetic sequence in which a1 � 5 and 1. __________________a12 � 38.

2. Find the 31st term in the arithmetic sequence 2. __________________9.3, 9, 8.7, 8.4, . . . .

3. Find the sum of the first 23 terms in the arithmetic 3. __________________series 6 � 11 � 16 � 21 � . . . .

4. Find the fifth term in the geometric sequence 4. __________________�10, �40, �160, . . . .

5. Find the sum of the first 10 terms in the geometric 5. __________________series 3 � 6 � 12 � 24 � . . . .

6. Form a sequence that has two geometric means 6. __________________between 9 and �13�.

7. Find limn→∞

�n2

2n�

2


8. Find the sum of the series �112� � �12� � 3 � . . . or state 8. __________________

that the sum does not exist.



10. �21

0� � �

21

2� � �

21

4� � . . . 10. __________________

11. �211� � �22

2� � �23

3� � . . . 11. __________________

Chapter

12


12. Write �7

k�43k in expanded form and then find the sum. 12. __________________

13. Express the series �12� 2� � �2 4

� 3� � �36� 4� � . . . � �81

�69� using 13. __________________

sigma notation.

14. Use the Binomial Theorem to expand (2p � 1)4. 14. __________________

15. Find the fourth term in the expansion of (2x � 3y)4. 15. __________________

16. Use the first f ive terms of the sine series sin x � 16. __________________x � �x3

3

!� � �x55

!� � �x77

!� � �x99

!� � . . . to f ind the value of

sin ��5� to four decimal places.

17. Find ln (�58) to four decimal places. 17. __________________

18. Find the first three iterates of the function ƒ(z) � 2z for 18. __________________z0 � 1 � 4i.

19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � i and z0 � 1.

20. Use mathematical induction to prove that 1 � 5 � 9 � . . . � 4n � 3 � n(2n � 1). Write your proof 20. __________________on a separate piece of paper.

Bonus Find the sum of the coefficients of the expansion Bonus: __________________of (x � y)5.

Chapter 12 Test, Form 2C (continued)


12


Chapter 12 Open-Ended Assessment

NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concisesolution to each problem. Be sure to include all relevant drawingsand justify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.

1. a. Write a word problem that involves an arithmetic sequence.Write the sequence and solve the problem. Tell what the answerrepresents.

b. Find the common difference and write the nth term of thearithmetic sequence in part a.

c. Find the sum of the first 12 terms of the arithmetic sequence inpart a. Explain in your own words why the formula for the sum ofthe first n terms of an arithmetic series works.

d. Does the related arithmetic series converge? Why or why not?

2. a. Write a word problem that involves a geometric sequence. Writethe sequence and solve the problem. Tell what the answerrepresents.

b. Find the common ratio and write the nth term of the geometricsequence in part a.

c. Find the sum of the first 11 terms of the sequence in part a.

d. Describe in your own words a test to determine whether a geometric series converges. Does the geometric series in part a converge?

3. a. Explain in your own words how to use mathematical induction toprove that a statement is true for all positive integers.

b. Use mathematical induction to prove that the sum of the first n terms of a geometric series is given by the formula

Sn � �a1

1�

�

ar1r

n

�, where r 1.

4. Find the fourth term in the expansion of ��

y2

x��

�yx��

6.

Chapter

12


1. Find the 20th term in the arithmetic sequence 1. __________________15, 21, 27, . . . .

2. Find the sum of the first 25 terms in the arithmetic 2. __________________series 11 � 14 � 17 � 20 � . . . .

3. Find the 12th term in the geometric sequence 3. __________________2�4, 2�3, 2�2, . . . .

4. Find the sum of the first 10 terms in the geometric 4. __________________series 2 � 6 � 18 � 54 � . . . .

5. Write a sequence that has two geometric means 5. __________________between 64 and �8.

6. Find limn→∞

�2n

n2

2

�

�

31n

� or state that the limit 6. __________________

does not exist.

7. Find the sum of the series �18� � �14� � �12� � . . . or state 7. __________________that the sum does not exist.



9. 5 � �15�

2

2� � �1 �52

3

� 3� � �1 � 25�

4

3 � 4� � . . . 9. __________________

10. �222� � �23

3� � �24

4� � �25

5� � . . . 10. __________________

Chapter 12 Mid-Chapter Test (Lessons 12-1 through 12-4)


12

1. Find the 11th term in the arithmetic sequence 1. __________________�3� � �5�, 0, ��3� � �5�, . . . .

2. Find n for the sequence for which an � 19, a1 � �13, 2. __________________and d � 2.

3. Find the sum of the first 17 terms in the arithmetic 3. __________________series 4.5 � 4.7 � 4.9 � . . . .

4. Find the fifth term in the geometric sequence for which 4. __________________a3 � �5� and r � 3.

5. Find the sum of the first six terms in the geometric 5. __________________series 1 � 1.5 � 2.25 � . . . .

6. Write a sequence that has one geometric mean 6. __________________between �13� and �2

57�.

Chapter 12, Quiz B (Lessons 12-3 and 12-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 12, Quiz A (Lessons 12-1 and 12-2)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

12

Chapter

12Find each limit, or state that the limit does not exist. 1. __________________

1. limn→∞

�2n

32n�

4

5� 2. lim

n→∞�(2n �

21n)(

2

n � 2)� 3. lim

n→∞�nn

2�

�

14

�2. __________________

3. __________________

Find the sum of each series, or state that the sum 4. __________________does not exist.

4. �12� � �14� � �18� � �116� � . . . 5. ��35� � 1 � �53� � . . . 5. __________________

6. Write the repeating decimal 0.4�5� as a fraction. 6. __________________

Determine whether each series is convergent or divergent.7. 0.002 � 0.02 � 0.2 � . . . 7. __________________

8. �58� � �59� � �150� � �1

51� � . . . 8. __________________

9. �11� 2� � �2

1� 3� � �3

1� 4� � �4

1� 5� � . . . 9. __________________

10. �12� 2� � �2

3� 3� � �3

4� 4� � . . . 10. __________________

1. Write �4

n�2�2n�1 � �12�� in expanded form and then find the sum. 1. __________________

2. Express the series �1861� � �2

87� � �49� � . . . using sigma notation. 2. __________________

3. Express the series 1 � 2 � 3 � 4 � 5 � 6 � . . . � 199 � 200 3. __________________using sigma notation.

4. Use the Binomial Theorem to expand (3a � d)4. 4. __________________

5. Use the first five terms of the exponential series 5. __________________ex � 1 � x � �x2

2

!� � �x33

!� � �x44

!� � . . . to approximate e4.1

to the nearest hundredth.

6. Use the first five terms of the trigonometric series 6. __________________sin x � x � �x3

3

!� � �x55

!� � �x77

!� � �x99

!� � . . . to approximate sin �

�3� to four decimal places.

Chapter 12, Quiz D (Lessons 12-8 and 12-9)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 12, Quiz C (Lessons 12-5 and 12-6)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

12

Chapter

121. Find the first four iterates of the function ƒ(x) � �1

10� x � 1 1. __________________

for x0 � 1.

2. Find the first three iterates of the function ƒ(z) � 2z � i 2. __________________for z0 � 3 � i.

3. Find the first three iterates of the function ƒ(z) � z2 � c 3. __________________for c � �1 � 2i and z0 � i.

4. Use mathematical induction to prove that 4. __________________1 � 3 � 5 � . . . � (2n � 1) � n2. Write your proof on a separate piece of paper.

5. Use mathematical induction to prove that 5. __________________5 � 11 � 17 � . . . � (6n � 1) � n(3n � 2). Write your proof on a separate piece of paper.


Chapter 12 SAT and ACT Practice


12After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

Multiple Choice1. In a basket of 80 apples, exactly 4 are

rotten. What percent of the apples arenot rotten?A 4%B 5%C 20%D 95%E 96%

2. Which grade had the largest percentincrease in the number of studentsfrom 1999 to 2000?

A 8B 9C 10D 11E 12

3. Find the length of a chord of a circle if the chord is 6 units from the center andthe length of the radius is 10 units.A 4B 8C 16D 2�3�4�E 4�3�4�

4. A chord of length 16 is 4 units fromthe center of a circle. Find the diameter.A 2�5�B 4�5�C 8�5�D 4�3�E 8�3�

5. If x � y � 4 and 2x � y � 5, then x � 2y �A 1B 2C 4D 5E 6

6. If �3kx15

�k

36� � 1 and x � 4, then k �

A 2B 3C 4D 8E 12

7. If 12% of a class of 25 students do nothave pets, how many students in theclass do have pets?A 3B 12C 13D 20E 22

8. In a senior class there are 400 boysand 500 girls. If 60% of the boys and50% of the girls live within 1 mile ofschool, what percent of the seniors donot live within 1 mile of school?A about 45.6%B about 54.4%C about 55.5%D about 44.4%E about 61.1%

9. In �ABC below, if BC � BA, which ofthe following is true?A x � yB y � zC y � xD y � xE z � x

Grade 8 9 10 11 121999 60 55 65 62 602000 80 62 72 72 70


Chapter 12 SAT and ACT Practice (continued)


1210. Which is the measure of each angle of

a regular polygon with r sides?A �1r�(360°)B (r � 2)180°C 360°D �1r�(r � 2)180°E 60°

11. A tractor is originally priced at $7000.The price is reduced by 20% and thenraised by 5%. What is the net reduction in price?A $5950 B $5880C $1400 D $1120E $1050

12. The original price of a camera provided a profit of 30% above thedealer’s cost. The dealer sets a newprice of $195, a 25% increase abovethe original price. What is the dealer’s cost?A $243.75 B $202.80C $156.00 D $120.00E None of these

13. Segments of the lines x � 4, x � 9,y � �5, and y � 4 form a rectangle.What is the area of this rectangle insquare units?A 6 B 10C 18 D 20E 45

14. In the figure below, the coordinates ofA are (4, 0) and of C are (15, 0). Findthe area of �ABC if the equation ofAB�� is 2x � y � 8.A 100 units2

B 121 units2

C 132 units2

D 144 units2

E 169 units2

15. In 1998, Bob earned $2800. In 1999,his earnings increased by 15%. In2000, his earnings decreased by 15%from his earnings in 1999. What werehis earnings in 2000?A $2800.00 B $2380.98C $2381.15 D $2737.00E None of these

16. Londa is paid a 15% commission onall sales, plus $8.50 per hour. Oneweek, her sales were $6821.29. Howmany hours did she work to earn$1371.69?A 68.3 B 41.0C 120.4 D 28.2E None of these

17–18. Quantitative ComparisonA if the quantity in Column A is

greaterB if the quantity in Column B is

greaterC if the two quantities are equalD if the relationship cannot be

determined from the informationgiven

Column A Column B

17. x > 0

18. One day, 90% of the girls and 80% ofthe boys were present in class.

19. Grid-In How many dollars must beinvested at a simple-interest rate of7.2% to earn $1440 in interest in 5 years?

20. Grid-In Seventy-two is 150% ofwhat number?

0.75xx plus an increase

of 75% of x

Number ofboys absent

Number ofgirls absent


Chapter 12 Cumulative Review (Chapters 1-12)

NAME _____________________________ DATE _______________ PERIOD ________

1. Solve the system by using a matrix equation. 1. __________________2x � 3y � 11y � 12 � x

2. Determine whether ƒ(x) � �xx2 �

�35x� is continuous at x � 3. 2. __________________

Justify your response using the continuity test.

3. Determine the binomial factors of 2x3 � x2 � 13x � 6. 3. __________________

4. Write an equation of the sine function with amplitude 1, 4. __________________period �23

��, phase shift �1�5�, and vertical shift 2.

5. Find the distance between the parallel lines 2x � 5y � 10 5. __________________and 2x � 5y � �5.

6. A 300-newton force and a 500-newton force act on the 6. __________________same object. The angle between the forces measures 95°.Find the magnitude and direction of the resultant force.

7. Find the product 4�cos �23�� i sin �23

�� 3�cos �56�� i sin �56

��. 7. __________________Then write the result in rectangular form.

8. Write the equation of the ellipse 6x2 � 9y2 � 54 after a 8. __________________rotation of 45° about the origin.

9. If $1500 is invested in an account bearing 8.5% interest 9. __________________compounded continuously, find the balance of the accountafter 18 months.

10. Express the series �33� � �65� � �97� � �192� � . . . � �32

01� using 10. __________________

sigma notation.

Chapter

12

BLANK


Trigonometry Semester Test

NAME _____________________________ DATE _______________ PERIOD ________


1. Solve 2x�1 � 17.6. Round your answer to the nearest hundredth. 1. ________A. 1.99 B. 3.14 C. �2.45 D. �3.84

2. Simplify i29 � i20. 2. ________A. 0 B. �2 C. �1 � i D. �2i

3. Find the third iterate of the function ƒ(x) � 2x � 1, if the initial 3. ________value is x0 � 3.A. 15 B. 31 C. 7 D. 12

4. Find the nineteenth term in the arithmetic sequence 10, 7, 4, 1, .... 4. ________A. 102 B. 0 C. �47 D. �44

5. If tan � � � �14� and � has its terminal side in Quandrant II, find the 5. ________exact value of sin �.A. �4 B. ��1�7� C. 2 D. �

�11�77�

�

6. Which expression is equivalent to tan ��2�

� ��? 6. ________

A. sin � B. cot � C. �cos � D. sec �

7. Find the ordered pair that represents the vector from A(1, �2) 7. ________to B(2, 3).A. 2, 3� B. 1, 2� C. 1, 5� D. 3, 1�

8. Which series is divergent? 8. ________A. 1 � �

21

3� � �

31

3� � �

41

3� � . . . B. 1 � �1

2� � �1

3� � . . .

C. 1 � �12� � �14� � �18� � . . . D. 2 � �23� � �29� � �227� � . . .

9. Which represents the graph of (3, 45°)? 9. ________A. B.

C. D.


10. Express x�29� y�

13� using radicals. 10. ________

A. �3

x�2y� B. �9

x�2y� C. �9

x�2y�3� �3

x�3y�

11. Which expression is equivalent to sin2 � for all values of �? 11. ________A. sin ��

�2� � �� B. 1 C. 2 sin � cos � D. 1 � cos2 �

12. Find the inner product of v� and w� if v� � 1, 2, 0� and w� � 3, �2, 1�. 12. ________A. 3 B. 2 C. �1 D. 1

13. Simplify (1 � i)(�2 � 2i). 13. ________A. 3 � 2i B. 2 � i C. �2 � 3i D. �4

14. Which equation is a trigonometric identity? 14. ________A. sin 2� � sin � cos � B. tan � � �s

cions��

C. tan 2� � �1

2�

ttaann

�2 �

� D. cos 2� � 4 cos2 � � 1

15. Write MN� as the sum of unit vectors for M(�11, 6, �7) and 15. ________N(4, �3, �15).A. �7i� � 3j� � 22k�� B. 15i� � 9j� � 8k��

C. �15i� � 9j�� 8k�� D. 7i� � 9j� � 8k��

16. Express �5

x�25�y�2� using rational exponents. 16. ________

A. x5y�52� B. x2y2 C. x5y�5

2� D. x�

12�y�5

2�

17. Express (x2 y3)�23� using radicals. 17. ________

A. �2

x�4y�6� B. �2

x�6y�9� C. �3

x�2y�3� D. �3

x�4y�6�

18. Which polar equation represents a rose? 18. ________A. r � 3� B. r � 3 � 3 sin �C. r � 3 cos 2� D. r2 � 4 cos 2�

19. Use a sum or difference identity to find the exact value of cos 105°. 19. ________

A. ��42�� B. �

�2� �4

�6�� C. �

�2� �4

�6�� D. �

�2� �

2�3�

�

Trigonometry Semester Test (continued)

NAME _____________________________ DATE _______________ PERIOD ________



NAME _____________________________ DATE _______________ PERIOD ________

20. Write the equation of the line 2x � 3y � 6 in parametric form. 20. ________

A. x � t; y � ��23�t � 2 B. x � 2t; y � 2t � 3

C. x � t; y � ��32�t � 2 D. x � 3t; y � t � 2

21. Find the sum of the geometric series �13� � �19� � �217� � . . . . 21. ________

A. �27� B. �12� C. �23� D. 1

22. Which is the graph of the equation � � �34��? 22. ________

A. B.

C. D.

23. Solve log6 x � 2. 23. ________A. 3 B. 36 C. 12 D. 4

24. Express 0.3 � 0.03 � 0.003 � . . . using sigma notation. 24. ________

A. �∞

k�1

3 � 10�k B. �∞

k�1

0.3 � 10�k

C. �∞

k�0

3 � 10�k D. �∞

k�1

3 � 10�2k

25. Andre kicks a soccer ball with an initial velocity of 48 feet per 25. ________second at an angle of 17° with the horizontal. After 0.35 second,what is the height of the ball?A. 16.07 ft B. 4.91 ft C. 14.11 ft D. 2.95 ft


26. Find v� � w� if v� � 2, �4, 6� and w� � 2, 1, �1�. 26. __________________

27. Write an equation in slope-intercept form of the line whose 27. __________________parametric equations are x � 4t � 3 and y � �2t � 7.

28. Use a calculator to find antiln (�0.23) to the nearest 28. __________________hundredth.

29. Find the first four terms of the geometric sequence for 29. __________________which a9 � 6561 and r � 3.

30. Find the polar coordinates of the point with rectangular 30. __________________coordinates (2, 2).

31. Find the rectangular coordinates of the point with polar 31. __________________coordinates ��2�, �

34��.

32. Express cos 840° as a trigonometric function of an angle 32. __________________in Quadrant II.

33. Solve 2 sin2 x � sin x � 0 for principle values of x. 33. __________________Express in degrees.

34. Find the distance between the point P(1, 0) and the line 34. __________________with equation 2x � 3y � �2.

35. If cos � � �34� and � has its terminal side in Quadrant I, find 35. __________________the exact value of sin �.

36. Express 2 � 4 � 6 � 8 � 10 using sigma notation. 36. __________________

37. Find limn→∞

�(3n � 2

n)2(n � 5)�. 37. __________________

38. Use a calculator to evaluate 3�3

�� to the nearest 38. __________________ten-thousandth.


NAME _____________________________ DATE _______________ PERIOD ________



NAME _____________________________ DATE _______________ PERIOD ________

39. Evaluate ��19��

12�

. 39. __________________

40. Use the Binomial Theorem to find the fourth term in 40. __________________the expansion of (2c � 3d)8.

41. Find the ordered triple that represents the vector 41. __________________from C(2, 0, 1) to D(6, 4, 3).

42. Find the distance between the lines with equations 42. __________________3x � 2y � 1 � 0 and 3x � 2y � 4 � 0.

43. Find an ordered pair to represent u� in 43. __________________u� � 2w� � v� if w� � 2, 3� and v� � 5, 5�.

44. Write the rectangular equation y � 1 in polar form. 44. __________________

45. Write the polar equation � � ��2� in rectangular form. 45. __________________

46. If CD� is a vector from C(1, 2, �1) to D(2, 3, 2), find 46. __________________the magnitude of CD�.

47. Are the vectors 2, �2, 1� and 3, 2, �2� perpendicular? 47. __________________Write yes or no.

48. Solve 235 � 4e0.35t. Round your answer to the 48. __________________nearest hundredth.


50. Graph the equation y � 3x�1. 50.

BLANK


Trigonometry Final Test

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.

1. Given ƒ(x) � 2x � 4 and g(x) � x3 � x � 2, find ( ƒ � g)(x). 1. ________A. x3 � x � 2 B. x3 � 3x � 6C. �x3 � 3x � 2 D. �x3 � x � 2

2. Describe the graph of ƒ(x) � �xx2 �

�255�. 2. ________

A. The graph has infinite discontinuity.B. The graph has jump discontinuity.C. The graph has point discontinuity.D. The graph is continuous.

3. Choose the graph of the relation whose inverse is a function. 3. ________A. B. C. D.

4. Which is the graph of the system? x 0 4. ________y 02x � y � 2

A. B. C. D.

5. For which measures does � ABC have no solution? 5. ________A. A � 60°, a � 8, b � 8B. A � 70°, b � 10, c � 10C. A � 45°, a � 1, b � 20D. A � 60°, B � 30°, c � 2


6. Choose the amplitude, period, and phase shift of the function 6. ________y � �13� cos (2x � �).

A. �13�, �, ��2� B. �13�, �, � C. 3, �, ��2� D. �13�, �, �4��

7. Given ƒ(x) � x � 1 and g(x) � x2 � 2, find [ g ° ƒ ](x). 7. ________A. x2 � 2x � 3 B. x2 � 2x � 1 C. x2 � 3 D. x3 � x2 � 2x � 2

8. Find one positive angle and one negative angle that are coterminal 8. ________with the angle �

34��.

A. �54��, ��

114

�� B. �

23��, ��

54�� C. �

114

��, � �

�4� D. �

114

��, ��

54��

9. Simplify �1 �si

cno�s2 ��. 9. ________

A. �csoins

�� B. tan � C. sin � D. cos �

10. Find an ordered triple to represent u� in u� � 2v� � 3w� if v� � �1, 0, 2� 10. ________and w� � 2, 3, 1�.A. 2, 3, 1� B. 4, 3, 2� C. 3, 1, �2� D. 4, 9, 7�

11. Write log7 49 � 2 in exponential form. 11. ________A. 27 � 49 B. 72 � 49

C. 7�12

�� 49 D. 49

�12

�� 7

12. Write parametric equations of �4x � 5y � 10. 12. ________A. x � t; y � �45�t � 2 B. x � t; y � 4t � 10C. x � t; y � �4t � 5 D. x � t; y � �54�t � 2

13. Describe the transformation that relates the graph of y � 100x3 13. ________to the parent graph y � x3.A. The parent graph is ref lected over the line y � x.B. The parent graph is stretched horizontally.C. The parent graph is stretched vertically.D. The parent graph is translated horizontally right 1 unit.

14. Identify the polar form of the linear equation �3�x � y � 4. 14. ________A. 4 � r cos (� � 30°) B. 2 � r cos (� � 30°)C. 3 � r sin (� � 70°) D. 2 � r cos (� � 42°)

Trigonometry Final Test (continued)

NAME _____________________________ DATE _______________ PERIOD ________



NAME _____________________________ DATE _______________ PERIOD ________

15. Which function is an even function? 15. ________A. y � x4 � x2 B. y � x3

C. y � x4 � x5 D. y � 2x4 � 3x

16. Find the area of � ABC to the nearest tenth if B � 30°, C � 70°, 16. ________and a � 10.A. 15.2 units2 B. 30.3 units2

C. 19.2 units2 D. 23.9 units2

17. The line with equation 9x � y � 2 is perpendicular to the line that 17. ________passes throughA. (0, 0) and (1, 3). B. (1, 2) and (10, 3).C. (2, �5) and (3, 4). D. (10, 4) and (1, 5).

18. Find the 17th term of the arithmetic sequence �13�, 1, �53�, . . . . 18. ________

A. 9 B. �137� C. 11 D. 7

19. Solve 2x2 � 4x � 1 � 0 by using the Quadratic Formula. 19. ________

A. 1 ��22�� B. �1 �

�22�� C. 2, 4 D. �1 �6�

20. What are the dimensions of matrix AB if A is a 2 � 3 matrix 20. ________and B is a 3 � 7 matrix?A. 7 � 2 B. 3 � 3 C. 3 � 2 D. 2 � 7

21. Find the polar coordinates of the point whose rectangular 21. ________coordinates are (�1, 1).

A. ��2�, ��4�� B. ��2�, ��

34��

C. ��2�, � ��4�� D. ��2�, �

34��

22. Find the fifteenth term of the geometric sequence �53�, �56�, �152�, . . . . 22. ________

A. �1965,608� B. �49,

5152� C. �98,

5304� D. �2

1�

23. Find the magnitude of AB� from A(2, 4, 0) to B(1, 2, 2). 23. ________A. �7� B. 2�3� C. 3 D. �5�


24. Choose the graph of y � csc x on the interval 0 � x � 2�. 24. ________A. B.

C. D.

25. Without graphing, describe the end behavior of y � �x4 � 2x3 � x2 � 1. 25. ________A. y → ∞ as x → ∞; y → ∞ as x → �∞B. y →�∞ as x → ∞; y → �∞ as x → �∞C. y →�∞ as x → ∞; y → ∞ as x → �∞D. y → ∞ as x → ∞; y → �∞ as x → �∞

26. Which expression is equivalent to sin (90° � �)? 26. ________A. sin � B. cos � C. tan � D. sec �

27. Describe the transformation that relates the graph of y � ��x� �� 3� 27. ________to the parent graph y � �x�.A. The parent graph is reflected over the x-axis and translated left 3 units.B. The parent graph is compressed vertically and translated right 3 units.C. The parent graph is reflected over the x-axis and translated up 3 units.D. The parent graph is reflected over the y-axis and translated left 3 units.

28. Express cos 854° as a function of an angle in Quandrant II. 28. ________A. tan 134° B. sin 64° C. cos 134° D. �cos 134°

29. Express �3

2�7�a�9b�2� by using rational exponents.

A. 3a3b�32� B. 9a6b�23

�C. 27a3b

�23

�D. 3a3b

�23

�29. ___________________________________

30. Identify the equation of the tangent function with period 2�, 30. ________phase shift �

�2�, and vertical shift �5.

A. y � tan (x � �) � 5 B. y � tan ��12�x � ��4�� 5

C. y � �5 tan ��12�x � ��4�� D. y � tan �2x � �

�2��


NAME _____________________________ DATE _______________ PERIOD ________



NAME _____________________________ DATE _______________ PERIOD ________

31. Given ƒ(x) � x3 � 2x, find ƒ(�2). 31. __________________

32. Find the area of a sector if the central angle measures 32. __________________35° and the radius of the circle is 45 centimeters. Round to the nearest tenth.

33. Find A � B if A � � and B � � . 33. __________________

34. Solve the system of equations algebraically. 34. __________________3x � 2y � 3�2x � 5y � 17

35. Find the inverse of y � (x � 1)3. 35. __________________

36. State the domain and range of the relation 36. __________________{(�2, 5), (0, 3), (4, 5), (9, �3)}. Then state whether the relation is a function.

37. Graph y � �|x � 1| � 2. 37.

38. Find the sum of the infinite geometric 38. __________________series 35 � 7 � 1.4 � . . . .

39. Find AB if A � � and B � � . 39. __________________

40. Simplify sin � csc � (sin2 � � cos2 �). 40. __________________

41. A football is kicked with an initial velocity of 38 feet per 41. __________________second at an angle of 32° to the horizontal. How far has the football traveled horizontally after 0.25 second? Round your answer to the nearest tenth.

42. Find the sum of the first 23 terms of the arithmetic 42. __________________series �10 � �3 � 4 � . . . .

23

3�1

�20

31

�14

22

33

1�2


43. Graph the polar equation � � ��3�. 43.

44. During one year, the cost of tuition, room, and board at a 44. __________________state university increased 6%. If the cost continues to increase at a rate of 6% per year, how long will it take for the cost of tuition, room, and board to triple? Round your answer to the nearest year.

45. Name four different pairs 45. __________________of polar coordinates thatrepresent the point A.

46. Evaluate log7 15 to four decimal places. 46. __________________

47. Use a calculator to approximate the value of sec (�137°) 47. __________________to four decimal places.

48. Find the sum of the first 13 terms of the geometric series 48. __________________2 � 6 � 18 � . . . .

49. Find the exact value of sin 105°. 49. __________________

50. Find the multiplicative inverse of � . 50. __________________

51. Find the value of � �. 51. __________________�1

423

21

52


NAME _____________________________ DATE _______________ PERIOD ________



NAME _____________________________ DATE _______________ PERIOD ________

52. The vector u� has a magnitude of 4.5 centimeters and a 52. __________________direction of 56°. Find the magnitude of its horizontal component. Round to the nearest hundredth.

53. Graph the equation y � 2 sin (2x � �). 53.

54. Write 3�2 � �19� in logarithmic form. 54. __________________

55. Use a sum or difference identity to f ind the value of 55. __________________cos 285°.

56. Solve e�1x

�� 5. Round your answer to the nearest hundredth. 56. __________________

57. Write MN� as a sum of unit vectors for M(2, 1, 0) 57. __________________and N(5, �3, 2).

58. Express 3 � 5 � 9 � 17 � . . . � 513 using sigma notation. 58. __________________

59. Express 2 � 3i in polar form. Express your answer in 59. __________________radians to the nearest hundredth.

60. Find the third iterate of the function ƒ(z) � 2z � 1, if the 60. __________________initial value is z0 � 1 � i.

61. Find the rectangular coordinates of the point whose polar 61. __________________

coordinates are �2�2�, �54��.

62. Find the discriminant of w2 � 4w � 5 � 0 and describe the 62. __________________nature of the roots of the equation.

63. Find the value of Cos�1 ��12�� in degrees. 63. __________________

64. State the degree and leading coefficient of the polynomial 64. __________________function ƒ(x) � �6x3 � 2x4 � 3x5 � 2.


65. Write an equation in slope-intercept form of the line with 65. __________________the given parametric equations. x � �3t � 5

y � 1 � t

66. Solve the system of equations. 66. __________________x � 5y � z � 12x � y � 2z � 2x � 3y � 4z � 6

67. Use the Remainder Theorem to find the remainder when 67. __________________2x3 � x2 � 2x is divided by x � 1.

68. Find the value of cos �Sin�1 ��23��. 68. __________________

69. Solve log5 x � log5 9 � log5 4. 69. __________________

70. Find the rational zero(s) of ƒ(x) � 3x3 � 2x2 � 6x � 4. 70. __________________

71. Find the number of possible positive real zeros of 71. __________________ƒ(x) � 3x4 � 2x3 � 3x2 � x � 3.

72. List all possible rational zeros of ƒ(x) � 3x3 � bx2 � cx � 2, 72. __________________where b and c are integers.

73. Given a central angle of 87°, f ind the length of its 73. __________________intercepted arc in a circle of radius 19 centimeters.Round to the nearest tenth.

74. Solve � ABC if A � 27.2°, B � 76.1°, and a � 31.2. Round 74. __________________to the nearest tenth.

75. Solve � ABC if A � 40°, b � 10.2, and c � 5.7. Round to 75. __________________the nearest tenth.


NAME _____________________________ DATE _______________ PERIOD ________

© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts

SAT and ACT Practice Answer Sheet(10 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

0 0 0

.. ./ /

.

99 9 9

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1


SAT and ACT Practice Answer Sheet(20 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

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99 9 987654321

87654321

87654321

87654321

Answers (Lesson 12-1)


© G

lenc

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1A

dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

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12-1 n

2�

n�

1; 5

1, 7

0, 9

25 � 2

3 � 2 1;

n2

�n

1 � 21 � 2

n2

�n

� 1

1 � 21 � 2

Qu

ad

ratic

Fo

rmu

las

for

Se

qu

en

ce

sA

n o

rdin

ary

arit

hm

etic

seq

uen

ce is

for

med

usi

ng

a ru

le s

uch

as

bn�

c. T

he

firs

t te

rm is

c,

bis

cal

led

the

com

mon

dif

fere

nce

, an

d n

take

s on

th

e va

lues

0, 1

, 2, 3

, an

d so

on

. Th

e va

lue

of t

erm

n�

1eq

ual

s b

(n�

1)�

cor

bn

�b

�c.

So,

th

e va

lue

of a

ter

m is

a

fun

ctio

n o

f th

e te

rm n

um

ber.

Som

e se

quen

ces

use

qu

adra

tic

fun

ctio

ns.

Am

eth

od c

alle

d fi

nit

ed

iffe

ren

ces

can

be

use

d to

fin

d th

e va

lues

of

the

term

s. N

otic

e w

hat

hap

pen

s w

hen

you

su

btra

ct t

wic

e as

sh

own

in t

his

tab

le.

nan

2�

bn�

c0

ca

�b

1a

�b

�c

3a�

b2a

24a

�2b

�c

5a�

b2a

39a

�3b

�c

7a�

b2a

416

a�

4b�

cA

sequ

ence

th

at y

ield

s a

com

mon

dif

fere

nce

aft

er t

wo

subt

ract

ion

sca

n b

e ge

ner

ated

by

a qu

adra

tic

expr

essi

on. F

or e

xam

ple,

th

ese

quen

ce 1

, 5, 1

2, 2

2, 3

5, .

. . g

ives

a c

omm

on d

iffe

ren

ce o

f 3

afte

rtw

o su

btra

ctio

ns.

Usi

ng

the

tabl

e ab

ove,

you

wri

te a

nd

solv

e th

ree

equ

atio

ns

to f

ind

the

gen

eral

ru

le. T

he

equ

atio

ns

are

1�

c,5

�a

�b

�c,

an

d 12

�4a

�2b

�c.

S

olve

eac

h p

rob

lem

.

1.R

efer

to

the

sequ

ence

in t

he

exam

ple

abov

e. S

olve

th

e sy

stem

of

equ

atio

ns

for

a, b

, an

d c

and

then

fin

d th

e qu

adra

tic

expr

essi

onfo

r th

e se

quen

ce.

Th

en w

rite

th

e n

ext

thre

e te

rms.

2.T

he

nu

mbe

r of

lin

e se

gmen

ts c

onn

ecti

ng

npo

ints

for

ms

the

sequ

ence

0, 0

, 1, 3

, 6, 1

0, .

. . ,

in w

hic

h n

is t

he

nu

mbe

r of

poi

nts

and

the

term

val

ue

is t

he

nu

mbe

r of

lin

e se

gmen

ts. W

hat

is t

he

com

mon

dif

fere

nce

aft

er t

he

seco

nd

subt

ract

ion

? F

ind

a qu

adra

tic

expr

essi

on f

or t

he

term

val

ue.

3.T

he

max

imu

m n

um

ber

of r

egio

ns

form

ed b

y n

chor

ds in

a c

ircl

efo

rms

the

sequ

ence

1, 2

, 4, 7

, 11,

16,

. . .

(A

chor

d is

a li

ne

segm

ent

join

ing

any

two

poin

ts o

n a

cir

cle.

) D

raw

cir

cles

to

illu

stra

te t

he

firs

t fo

ur

term

s of

th

e se

quen

ce.

Th

en f

ind

a qu

adra

tic

expr

essi

onfo

r th

e te

rm v

alu

e.

© G

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0A

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Mat

hem

atic

al C

once

pts

Ari

thm

etic

Se

qu

en

ce

s a

nd

Se

rie

sFi

nd

th

e n

ext

fou

r te

rms

in e

ach

ari

thm

etic

seq

uen

ce.

1.�

1.1,

0.6

, 2.3

, . .

.2.

16, 1

3, 1

0, .

. .3.

p, p

�2,

p�

4,. .

.4.

0, 5

.7, 7

.4, 9

.17,

4, 1

,�2

p�

6, p

�8,

p�

10, p

�12

For

exer

cise

s 4

–12,

ass

um

e th

at e

ach

seq

uen

ce o

r se

ries

is a

rith

met

ic.

4.F

ind

the

24th

ter

m in

th

e se

quen

ce f

or w

hic

h a

1�

�27

an

d d

�3.

42

5.F

ind

nfo

r th

e se

quen

ce f

or w

hic

h a

n�

27, a

1�

�12

, an

d d

�3.

14

6.F

ind

dfo

r th

e se

quen

ce f

or w

hic

h a

1�

�12

an

d a 23

�32

.2

7.W

hat

is t

he

firs

t te

rm in

th

e se

quen

ce f

or w

hic

h d

��

3 an

d a 6

�5?

20

8.W

hat

is t

he

firs

t te

rm in

th

e se

quen

ce f

or w

hic

h d

��

�1 3�an

d a 7

��

3?�

1

9.F

ind

the

6th

ter

m in

th

e se

quen

ce�

3�

�2�,

0, 3

��

2�, .

. . .

12�

4�2�

10.F

ind

the

45th

ter

m in

th

e se

quen

ce�

17,�

11,�

5, .

. ..

247

11.

Wri

te a

seq

uen

ce t

hat

has

th

ree

arit

hm

etic

mea

ns

betw

een

35

and

45.

35, 3

7.5,

40,

42.

5, 4

5

12.W

rite

a s

equ

ence

th

at h

as t

wo

arit

hm

etic

mea

ns

betw

een

�7

and

2.75

.�

7,�

3.75

,�0.

5, 2

.75

13.F

ind

the

sum

of

the

firs

t 13

ter

ms

in t

he

seri

es�

5�

1�

7�

. . .

�67

.40

3

14.F

ind

the

sum

of

the

firs

t 62

ter

ms

in t

he

seri

es�

23�

21.5

�20

�. .

. .

1410

.5

15.A

ud

itor

ium

Des

ign

Wak

efie

ld A

udi

tori

um

has

26

row

s, a

nd

the

firs

t ro

w h

as 2

2se

ats.

Th

e n

um

ber

of s

eats

in e

ach

row

incr

ease

s by

4 a

s yo

u m

ove

tow

ard

the

back

of t

he

audi

tori

um

. Wh

at is

th

e se

atin

g ca

paci

ty o

f th

is a

udi

tori

um

?18

72

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

12-1



© G

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4A

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Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

12-2

Se

qu

en

ce

s a

s F

un

ctio

ns

Age

omet

ric

seq

uen

ceca

n b

e de

fin

ed a

s a

fun

ctio

n w

hos

e do

mai

n

is t

he

set

of p

osit

ive

inte

gers

.n

�1

23

4...

↓↓

↓↓

↓f(

n)�

ar1

– 1

ar2

– 1

ar3

– 1

ar4

– 1

...In

th

e ex

erci

ses,

you

wil

l hav

e th

e op

port

un

ity

to e

xplo

re g

eom

etri

cse

quen

ces

from

a f

un

ctio

n a

nd

grap

hin

g po

int

of v

iew

.

Gra

ph

eac

h g

eom

etri

c se

qu

ence

for

n =

1, 2

, 3 a

nd

4.

1.f(

n)�

2n2.

f(n

)�(0

.5)n

3.f(

n)�

(–2)

n4.

f(n

)�(–

0.5)

n

5.D

escr

ibe

how

th

e gr

aph

of

a ge

omet

ric

sequ

ence

dep

ends

on

th

eco

mm

on r

atio

.r

�1:

gra

ph

rise

s to

the

rig

ht.

r�

– 1:

gra

ph

rise

s an

d f

alls

and

hig

h an

d lo

w p

oin

ts

mo

ve a

way

fro

m t

he n

-axi

s.0

�r

�1:

gra

ph

falls

to

the

rig

ht.

– 1�

r�

0: g

rap

h ri

ses

and

fal

ls a

nd h

igh

and

low

po

ints

ap

pro

ach

the

n-a

xis.

6.L

et f

(n)�

2n ,

wh

ere

n is

a p

osit

ive

inte

ger.

a.S

how

gra

phic

ally

th

at f

or a

ny

Mth

e gr

aph

of

f(n

) ri

ses

abov

e an

d st

ays

abov

e th

e h

oriz

onta

l lin

e y

�M

.b

.S

how

alg

ebra

ical

ly t

hat

for

an

y M

, th

ere

is a

pos

itiv

e in

tege

r N

such

th

at 2

n�

Mfo

r al

l n

�N

.C

hoo

se a

po

siti

ve in

teg

er N

�lo

g2M

.T

hen

2N�

2log

2M�

M.

© G

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Mat

hem

atic

al C

once

pts

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Ge

om

etr

ic S

eq

ue

nc

es

an

d S

eri

es

Det

erm

ine

the

com

mon

rat

io a

nd

fin

d t

he

nex

t th

ree

term

s of

eac

h g

eom

etri

cse

qu

ence

.

1.�

1, 2

,�4,

. . .

2.�

4,�

3,�

�9 4� , .

. .3.

12,�

18, 2

7, .

. .

�2;

8,�

16, 3

2�3 4� ;

��2 17 6�

,��8 61 4�

,��2 24 53 6�

��3 2� ;

��8 21 �

, �24 43 �

, ��72 89 �

For

exer

cise

s 4

–9, a

ssu

me

that

eac

h s

equ

ence

or

seri

es is

geo

met

ric.

4.F

ind

the

fift

h t

erm

of

the

sequ

ence

20,

0.2

, 0.0

02, .

. . .

0.00

0000

2

5.F

ind

the

nin

th t

erm

of

the

sequ

ence

�3�,

�3,

3�

3�, .

. . .

81�

3�

6.If

r�

2 an

d a 4

�28

, fin

d th

e fi

rst

term

of

the

sequ

ence

. � 27 �

7.F

ind

the

firs

t th

ree

term

s of

th

e se

quen

ce f

or w

hic

h a

4�

8.4

and

r�

4.0.

1312

5, 0

.525

, 2.1

8.F

ind

the

firs

t th

ree

term

s of

th

e se

quen

ce f

or w

hic

h a

6�

� 31 2�an

d r

��1 2� .

1, �1 2� ,

� 41 �

9.W

rite

a s

equ

ence

th

at h

as t

wo

geom

etri

c m

ean

s be

twee

n 2

an

d 0.

25.

2, 1

, 0.5

, 0.2

5

10.W

rite

a s

equ

ence

th

at h

as t

hre

e ge

omet

ric

mea

ns

betw

een

�32

an

d�

2.�

32, �

16,�

8,�

4,�

2

11.

Fin

d th

e su

m o

f th

e fi

rst

eigh

t te

rms

of t

he

seri

es �3 4�

�� 29 0�

�� 12 07 0�

�. .

. .

abo

ut 1

.843

51

12.F

ind

the

sum

of

the

firs

t 10

ter

ms

of t

he

seri

es�

3�

12�

48�

. . .

.62

9,14

5

13.P

opu

lati

on G

row

thA

city

of

100,

000

peop

le is

gro

win

g at

a r

ate

of

5.2%

per

yea

r. A

ssu

min

g th

is g

row

th r

ate

rem

ain

s co

nst

ant,

est

imat

e th

e po

pula

tion

of

the

city

5 y

ears

fro

m n

ow.

abo

ut 1

28,8

48

12-2



© G

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Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

12-3

So

lvin

g E

qu

atio

ns

Usi

ng

Se

qu

en

ce

sYo

u c

an u

se s

equ

ence

s to

sol

ve m

any

equ

atio

ns.

For

exa

mpl

e,

con

side

r x2

�x

– 1

�0.

You

can

pro

ceed

as

foll

ows.

x2�

x –

1�

0

x(x

�1)

�1

x�

Nex

t, d

efin

e th

e se

quen

ce: a

1�

0 an

d a n

�.

Th

e li

mit

of

the

sequ

ence

is a

sol

uti

on t

o th

e or

igin

al e

quat

ion

.

1.L

et a

1�

0 an

d a n

�.

a.W

rite

th

e fi

rst

five

ter

ms

of t

he

sequ

ence

. Do

not

sim

plif

y.

1

11

1__

___

__

___

____

____

___

____

____

____

____

_0,

1 +

0 ,

1 +

1 ,

1 +

1

,1

+

1__

___

____

____

___

1 +

11

+1

____

__1

+ 1

b.

Wri

te d

ecim

als

for

the

firs

t fi

ve t

erm

s of

th

e se

quen

ce.

0, 1

, 0.5

, 0.6

667,

0.6

c.U

se a

cal

cula

tor

to c

ompu

te a

6, a

7, a

8, a

nd

a 9. C

ompa

re a

9w

ith

the

posi

tive

sol

uti

on o

f x2

�x

– 1

�0

fou

nd

by u

sin

g th

e qu

adra

tic

form

ula

.0.

625,

0.6

154,

0.6

190,

0.6

176;

so

luti

on

by

qua

dra

tic

form

ula:

0.6

180

2.U

se t

he

met

hod

des

crib

ed a

bove

to

fin

d a

root

of

3x2

–2x

–3

�0.

–0.7

208

3.W

rite

a B

AS

IC p

rogr

am u

sin

g th

e pr

oced

ure

ou

tlin

ed a

bove

to

fin

d a

root

of

the

equ

atio

n 3

x2–

2x–

3�

0. I

n t

he

prog

ram

,

let

a 1�

0 an

d a n

�.

Ru

n t

he

prog

ram

. Com

pare

th

e

tim

e it

tak

es t

o ru

n t

he

prog

ram

to

the

tim

e it

tak

es t

o ev

alu

ate

the

term

s of

th

e se

quen

ce b

y u

sin

g a

calc

ula

tor.

10 D

IM A

[100

]20

A[1

] �

030

FO

R I

�2

TO

100

40 A

[I]

�3/

(3*A

[I �

1] �

2)50

NE

XT

I60

PR

INT

A[1

00]3

��

3an

�1�

2

1�

�1

� a

n�

1

1�

�1

� a

n�

1

1� 1

�x

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Mat

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once

pts

Infi

nite

Se

qu

en

ce

an

d S

eri

es

Fin

d e

ach

lim

it, o

r st

ate

that

th

e lim

it d

oes

not

exi

st a

nd

exp

lain

you

r re

ason

ing

.1.

lim

n→

∞� nn 22

��11

�2.

lim

n→

∞�4 3n n2 2� �

5 4n�

1� 34 �

3.li

mn

→∞�5n

62

n�1

�4.

lim

n→

∞�(n

�1 5)( n3 2n

�1)

�

do

es n

ot

exis

t; d

ivid

ing

by

n2 ,

� 53 �

we

find

lim

n→

∞, w

hich

sim

plif

ies

to �5 6� �

00�

��5 0�

as n

app

roac

hes

infin

ity.

Sin

ce t

his

frac

tio

n is

und

efin

ed, t

he

seq

uenc

e ha

s no

lim

it.

5.li

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lim

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oes

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��

lim n→

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�0,

but

as

n

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roac

hes

infin

ity,

nb

eco

mes

in

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sing

ly la

rge,

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the

se

que

nce

has

no li

mit

.

Wri

te e

ach

rep

eati

ng

dec

imal

as

a fr

acti

on.

7.0.

7�5�8.

0.5�9�

2��2 35 3�

� 21 76 �

Fin

d t

he

sum

of

each

infi

nit

e se

ries

, or

stat

e th

at t

he

sum

doe

s n

ot e

xist

an

d

exp

lain

you

r re

ason

ing

.

9.�2 5�

�� 26 5�

�� 11 28 5�

�. .

.10

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��7 15 6�

�. .

.

1d

oes

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t ex

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sin

ce r

��5 2� ,

�r��

1 an

d

the

seq

uenc

e in

crea

ses

wit

hout

lim

it.

11.

Ph

ysic

sA

ten

nis

bal

l is

drop

ped

from

a h

eigh

t of

55

feet

an

dbo

un

ces

�3 5�of

th

e di

stan

ce a

fter

eac

h f

all.

a.F

ind

the

firs

t se

ven

ter

ms

of t

he

infi

nit

e se

ries

rep

rese

nti

ng

the

vert

ical

dis

tan

ces

trav

eled

by

the

ball

.55

, 33,

33,

19.8

, 19.

8, 1

1.88

, 11.

88b

.W

hat

is t

he

tota

l ver

tica

l dis

tan

ce t

he

ball

tra

vels

bef

ore

com

ing

to r

est?

220

ft

5�

� n1 2��

� n6 �

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

12-3

© G

lenc

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w-H

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dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

12-4

Alt

ern

atin

g S

eri

es

Th

e se

ries

bel

ow is

cal

led

an a

lter

nat

ing

seri

es.

1 –

1�

1 –

1�

...

Th

e re

ason

is t

hat

th

e si

gns

of t

he

term

s al

tern

ate.

An

inte

rest

ing

ques

tion

is w

het

her

th

e se

ries

con

verg

es. I

n t

he

exer

cise

s, y

ou w

ill

hav

e an

opp

ortu

nit

y to

exp

lore

th

is s

erie

s an

d ot

her

s li

ke it

.

1.C

onsi

der

1 –

1�

1 –

1�

....

a.W

rite

an

arg

um

ent

that

su

gges

ts t

hat

th

e su

m is

1.

1�

1�

1�

1�

...�

1�

(–1

�1)

�(–

1�

1 )�

...�

1�

0�

0�

...�

1b

.W

rite

an

arg

um

ent

that

su

gges

ts t

hat

th

e su

m is

0.

1�

1�

1�

1�

...�

(1�

1)�

(1�

1)�

(1�

1)�

...�

0�

0�

0�

...�

0c.

Wri

te a

n a

rgu

men

t th

at s

ugg

ests

th

at t

her

e is

no

sum

. (H

int:

Con

side

r th

e se

quen

ce o

f pa

rtia

l su

ms.

)Le

t S

nb

e th

e n

th p

arti

al s

um.

The

n

S n�

�S

ince

1, 0

, 1, 0

, ...

has

no li

mit

, the

ori

gin

alse

ries

has

no

sum

.

If t

he

seri

es f

orm

ed b

y ta

kin

g th

e ab

solu

te v

alu

es o

f th

e te

rms

of a

give

n s

erie

s is

con

verg

ent,

th

en t

he

give

n s

erie

s is

sai

d to

be

abso

lute

ly c

onve

rgen

t. I

t ca

n b

e sh

own

th

at a

ny

abso

lute

ly

con

verg

ent

seri

es is

con

verg

ent.

2.M

ake

up

an a

lter

nat

ing

seri

es, o

ther

th

an a

geo

met

ric

seri

es w

ith

neg

ativ

e co

mm

on r

atio

, th

at h

as a

su

m. J

ust

ify

you

r an

swer

.S

amp

le a

nsw

er:

��

��

...is

co

nver

gen

t b

ecau

se

the

abso

lute

val

ue o

f ea

ch t

erm

is le

ss t

han

or

equa

l to

the

co

rres

po

ndin

g t

erm

in a

p-s

erie

sw

ith

p�

2.

1 � 161 � 9

1 � 41 � 1

1 if

nis

od

d.

0 if

nis

eve

n.

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once

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E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Co

nve

rge

nt

an

d D

ive

rge

nt

Se

rie

s

Use

th

e ra

tio

test

to

det

erm

ine

wh

eth

er e

ach

ser

ies

isco

nve

rgen

t or

div

erg

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conv

erg

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div

erg

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3.� 1

�

4 2�

3�

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8 �3

�4

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�5

��

. . .

conv

erg

ent

4.5

�� 35 3�

�� 55 3�

�� 75 3�

�. .

.

div

erg

ent

Use

th

e co

mp

aris

on t

est

to d

eter

min

e w

het

her

eac

h s

erie

s is

con

verg

ent

or d

iver

gen

t.5.

2�

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� 32 3��

� 42 3��

. . .

6.�5 2�

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�� 15 1�

�. .

.

conv

erg

ent

div

erg

ent

7.E

colo

gyA

lan

dfil

l is

leak

ing

a to

xic

chem

ical

. Six

mon

ths

afte

rth

e le

ak w

as d

etec

ted,

th

e ch

emic

al h

ad s

prea

d 12

50 m

eter

s fr

omth

e la

ndf

ill.

Aft

er o

ne

year

, th

e ch

emic

al h

ad s

prea

d 50

0 m

eter

sm

ore,

an

d by

th

e en

d of

18

mon

ths,

it h

ad r

each

ed a

n a

ddit

ion

al20

0 m

eter

s.a.

If t

his

pat

tern

con

tin

ues

, how

far

wil

l th

e ch

emic

al s

prea

dfr

om t

he

lan

dfil

l aft

er 3

yea

rs?

abo

ut 2

075

m

b.

Wil

l th

e ch

emic

al e

ver

reac

h t

he

grou

nds

of

a h

ospi

tal l

ocat

ed25

00 m

eter

s aw

ay f

rom

th

e la

ndf

ill?

Exp

lain

.N

o, t

he s

um o

f th

e in

finit

e se

ries

mo

del

ing

thi

s si

tuat

ion

is a

bo

ut 2

083.

Thu

s th

e ch

emic

al w

illsp

read

no

mo

re t

han

abo

ut 2

083

met

ers.

12-4





© G

lenc

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cGra

w-H

ill52

3A

dva

nced

Mat

hem

atic

al C

once

pts

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Str

ee

t N

etw

ork

s: F

ind

ing

All P

oss

ible

Ro

ute

sA

sect

ion

of

a ci

ty is

laid

ou

t in

squ

are

bloc

ks.

Goi

ng

nor

th f

rom

th

e in

ters

ecti

on o

f 1s

t A

ven

ue

and

1st

Str

eet,

th

e av

enu

es a

re

1st,

2n

d, 3

rd, a

nd

so o

n.

Goi

ng

east

, th

e st

reet

s ar

e n

um

bere

d in

th

e sa

me

way

.

Fac

tori

als

can

be

use

d to

fin

d th

e n

um

ber,

r(

e, n

), o

f di

ffer

ent

rou

tes

betw

een

tw

o in

ters

ecti

ons.

Th

e n

um

ber

of s

tree

ts g

oin

g ea

st is

e; t

he

nu

mbe

r of

ave

nu

es g

oin

gn

orth

is n

.

Th

e fo

llow

ing

prob

lem

s ex

amin

e th

e po

ssib

le r

oute

s fr

om o

ne

loca

tion

to

anot

her

. A

ssu

me

that

you

nev

er u

se a

rou

te t

hat

is

un

nec

essa

rily

lon

g. A

ssu

me

that

e�

1 an

d n

�1.

Sol

ve e

ach

pro

ble

m.

1.L

ist

all t

he

poss

ible

rou

tes

from

1st

Str

eet

and

1st

Ave

nu

e to

4t

h S

tree

t an

d 3r

d A

ven

ue.

Use

ord

ered

pai

rs t

o sh

ow t

he

rou

tes,

w

ith

str

eet

nu

mbe

rs f

irst

an

d av

enu

e n

um

bers

sec

ond.

Eac

h

rou

te m

ust

sta

rt a

t (1

, 1)

and

end

at (

4, 3

).(1

, 1) →

(2, 1

) → (3

, 1) →

(4, 1

) → (4

, 2) →

(4, 3

)(1

, 1) →

(2, 1

) → (3

, 1) →

(3, 2

) → (4

, 2) →

(4, 3

)(1

, 1) →

(2, 1

) → (3

, 1) →

(3, 2

) → (3

, 3) →

(4, 3

)(1

, 1) →

(2, 1

) → (2

, 2) →

(3, 2

) → (4

, 2) →

(4, 3

)(1

, 1) →

(2, 1

) → (2

, 2) →

(3, 2

) → (3

, 3) →

(4, 3

)(1

, 1) →

(2, 1

) → (2

, 2) →

(2, 3

) → (3

, 3) →

(4, 3

)(1

, 1) →

(1, 2

) → (2

, 2) →

(3, 2

) → (4

, 2) →

(4, 3

)(1

, 1) →

(1, 2

) → (2

, 2) →

(3, 2

) → (3

, 3) →

(4, 3

)(1

, 1) →

(1, 2

) → (2

, 2) →

(2, 3

) → (3

, 3) →

(4, 3

)(1

, 1) →

(1, 2

) → (1

, 3) →

(2, 3

) → (3

, 3) →

(4, 3

)2.

Use

the

form

ula

toco

mpu

teth

en

um

ber

ofro

ute

sfr

om (

1,1)

to(4

,3).

Th

ere

are

4 st

reet

s go

ing

east

an

d 3

aven

ues

goi

ng

nor

th.

�10

3.F

ind

the

nu

mbe

r of

rou

tes

from

1st

Str

eet

and

1st

Ave

nu

e to

7th

Str

eet

and

6th

Ave

nu

e.

�46

2(6

�5)

!�

6!5!

(3�

2)!

�3!

2!

Enr

ichm

ent

12-5

r(e,

n)

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e�

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�1)

!

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No

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th T

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Wri

te e

ach

exp

ress

ion

in e

xpan

ded

for

m a

nd

th

en f

ind

th

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m.

1.�5

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2.�4

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; 40

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26.

�∞ i�110

��1 2� �i

(2�

2)2

�(3

�2)

2�

10��1 2� �1

�10

��1 2� �2�

(4�

2)2

�(5

�2)

2 ; 1

410

��1 2� �3�

. . .

�10

��1 2� ��; 1

0

Exp

ress

eac

h s

erie

s u

sin

g s

igm

a n

otat

ion

.

7.3

�6

�9

�12

�15

8.6

�24

�12

0�

. . .

�40

,320

�5

n�

13n

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n�

3n

!

9.�1 1�

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24�

19�

14�

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1)

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n�

1� n1 2�

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4�

5n)

11.

Sa

vin

gsK

ath

ryn

sta

rted

sav

ing

quar

ters

in a

jar.

Sh

e be

gan

by p

utt

ing

two

quar

ters

in t

he

jar

the

firs

t da

y an

d th

en s

he

incr

ease

d th

e n

um

ber

of q

uar

ters

sh

e pu

t in

th

e ja

r by

on

e ad

diti

onal

qu

arte

r ea

ch s

ucc

essi

ve d

ay.

a.U

se s

igm

a n

otat

ion

to

repr

esen

t th

e to

tal n

um

ber

of q

uar

ters

Kat

hry

n h

ad a

fter

30

days

.

b.

Fin

d th

e su

m r

epre

sen

ted

in p

art

a.

495

qua

rter

s

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

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____

____

____

_ P

ER

IOD

____

____

12-5

�30

n�

1(n�

1)



© G

lenc

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w-H

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6A

dva

nced

Mat

hem

atic

al C

once

pts

Pa

tte

rns

in P

asc

al’s

Tri

an

gle

You

hav

e le

arn

ed t

hat

th

e co

effi

cien

ts in

th

e ex

pan

sion

of

(x

�y)

n

yiel

d a

nu

mbe

r py

ram

id c

alle

d P

asca

l’s t

rian

gle.

Row

1

Row

2

Row

3

Row

4

Row

5

Row

6

Row

7

As

man

y ro

ws

can

be

adde

d to

th

e bo

ttom

of

the

pyra

mid

as

you

nee

d.T

his

act

ivit

y ex

plor

es s

ome

of t

he

inte

rest

ing

prop

erti

es o

f th

is

fam

ous

nu

mbe

r py

ram

id.

1.P

ick

a ro

w o

f P

asca

l’s t

rian

gle.

a.W

hat

is t

he

sum

of

all t

he

nu

mbe

rs in

all

th

e ro

ws

abov

e th

e ro

w y

ou p

icke

d?S

ee s

tud

ents

’ wo

rk.

b.W

hat

is t

he

sum

of

all t

he

nu

mbe

rs in

th

e ro

w y

ou p

icke

d?S

ee s

tud

ents

’ wo

rk.

c.H

ow a

re y

our

answ

ers

for

part

s a

and

bre

late

d?T

he a

nsw

er f

or

par

t b

is 1

mo

re t

han

the

answ

er f

or

par

t a.

d.R

epea

t pa

rts

ath

rou

gh c

for

at le

ast

thre

e m

ore

row

s of

P

asca

l’s t

rian

gle.

Wh

at g

ener

aliz

atio

n s

eem

s to

be

tru

e?It

app

ears

that

the

sum

of t

he n

umbe

rs in

any

row

is 1

mor

eth

an th

e su

m o

f the

num

bers

in a

ll of

the

row

s ab

ove

it.e.

See

if y

ou c

an p

rove

you

r ge

ner

aliz

atio

n.

Sum

of n

umbe

rs in

row

n�

2n�

1 ; Th

e su

m o

f the

num

bers

in th

ero

ws

abov

e ro

w n

is 2

0+

21+

22+

. . .

+ 2n

�2 ,

whi

ch, b

y th

e fo

rmul

a fo

r the

sum

of a

geo

met

ric s

erie

s, is

2n

�1

�1.

2.P

ick

any

row

of

Pas

cal’s

tri

angl

e th

at c

omes

aft

er t

he

firs

t.a.

Sta

rtin

g at

th

e le

ft e

nd

of t

he

row

, fin

d th

e su

m o

f th

e od

d n

um

bere

d te

rms.

See

stu

den

ts’ w

ork

.b

.In

th

e sa

me

row

, fin

d th

e su

m o

f th

e ev

en n

um

bere

d te

rms.

See

stu

den

ts’ w

ork

.c.

How

do

the

sum

s in

par

ts a

an

d b

com

pare

?T

he s

ums

are

equa

l.d

.Rep

eat

part

s a

thro

ugh

cfo

r at

leas

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tem

ent

is t

rue

for

n�

4.

2.R

efer

to

the

diag

ram

s at

th

e ri

ght.

a.

How

man

y do

ts w

ould

th

ere

be in

th

e fo

urt

hdi

agra

m S

4in

th

e se

quen

ce?

16 b.

Des

crib

e a

met

hod

th

at y

ou c

an u

se t

o de

term

ine

the

nu

mbe

r of

dot

s in

th

e fi

fth

dia

gram

S5

base

d on

th

e n

um

ber

of d

ots

in t

he

fou

rth

dia

gram

, S4.

Ver

ify

you

r an

swer

by

con

stru

ctin

g th

e fi

fth

dia

gram

.

Ad

d 5

to

the

num

ber

of

do

ts in

S4.

S

5w

oul

d h

ave

16 �

5 o

r 21

do

ts.

c.F

ind

a fo

rmu

la t

hat

can

be

use

d to

com

pute

th

e n

um

ber

of d

ots

in t

he

nth

dia

gram

of

this

seq

uen

ce.

Use

mat

hem

atic

al in

duct

ion

to

prov

e yo

ur

form

ula

isco

rrec

t.S n

�5

n�

4Ve

rify

that

Sn

is tr

ue fo

r n

�1.

S

1�

5(1)

�4

or 1

. Ass

ume

S nis

true

fo

r n

�k.

Pro

ve it

is tr

ue fo

r n

�k

�1.

12-9

S k�

1�

S k�

5S k

�1

�5(

k�

1)�

4�

5k�

5�

4�

(5k

�4)

�5

�S k

�5

S1

S2

S3

© G

lenc

oe/M

cGra

w-H

ill53

4A

dva

nced

Mat

hem

atic

al C

once

pts

Ma

the

ma

tic

al In

du

ctio

nU

se m

ath

emat

ical

ind

uct

ion

to

pro

ve t

hat

eac

h p

rop

osit

ion

is v

alid

for

all p

osit

ive

inte

gra

l val

ues

of

n.

1.�1 3�

��2 3�

��3 3�

�. .

.�

�n 3��

�n(n

6�1)

�

Ste

p 1

: Ver

ify t

hat

the

form

ula

is v

alid

fo

r n

�1.

Sin

ce �1 3�

is t

he f

irst

ter

m in

the

sen

tenc

e an

d�1(

16�

1)�

��1 3� ,

the

form

ula

is v

alid

fo

r n

�1.

Ste

p 2

: Ass

ume

that

the

fo

rmul

a is

val

id f

or

n�

kan

d t

hen

pro

ve t

hat

it is

als

o v

alid

fo

r n

�k

�1.

Sk

⇒ �1 3�

��2 3�

��3 3�

�. .

.�

�k 3��

�k(k 6�

1)�

Sk�

1⇒

�1 3��

�2 3��

�3 3��

. . .

��k 3�

��k

� 31

��

�k(k

6�1)

��

�k� 3

1�

� ��(k

�1 6)(k

�2)

�

Ap

ply

the

ori

gin

al f

orm

ula

for

n�

k�

1.�k

�1(

k6�

1�

1)�

or �(k

�1 6)(k

�2)

�

Thu

s, if

the

fo

rmul

a is

val

id f

or

n�

k, it

is a

lso

val

id f

or

n�

k�

1.S

ince

the

fo

rmul

a is

val

id f

or

n�

1, it

is a

lso

val

id f

or

n�

2, n

�3,

and

so o

n. T

hat

is, t

he f

orm

ula

is v

alid

fo

r al

l po

siti

ve in

teg

ral v

alue

s o

f n

.2.

5n�

3 is

div

isib

le b

y 4.

Ste

p 1

: Ver

ify t

hat

Sn

is v

alid

fo

r n

�1.

S1

⇒51

�3

�8.

Sin

ce 8

is d

ivis

ible

by

4, S

nis

val

id f

or

n�

1.S

tep

2: A

ssum

e th

at S

nis

val

id f

or

n�

kan

d t

hen

pro

ve t

hat

it is

val

id

for

n�

k�

1.S

k⇒

5k�

3�

4rfo

r so

me

inte

ger

r

Sk�

1⇒

5k�1

�3

�4t

for

som

e in

teg

er t

5k�

3�

4r5(

5k�

3)�

5(4r

)5k�

1�

15�

20r

5k�1

�3

�20

r�

125k�

1�

3�

4(5

r�

3)Le

t t

�5r

�3,

an

inte

ger

. The

n 5k�

1�

3�

4t.

Thu

s, if

Sk

is v

alid

, the

n S

k�1 is

als

o v

alid

. Sin

ce S

nis

val

id f

or

n�

1, it

is a

lso

val

id f

or

n�

2, n

�3,

and

so

on.

Hen

ce, 5

n�

3 is

div

isib

le b

y 4

for

all p

osi

tive

inte

gra

l val

ues

of

n.

k(k

�1)

�2(

k�

1)�

��

6

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

12-9


Page 537

1. D

2. C

3. A

4. B

5. A

6. A

7. A

8. B

9. C

10. D

11. D

Page 538

12. C

13. B

14. A

15. B

16. C

17. C

18. B

19. D

20. D

Bonus: C

Page 539

1. D

2. B

3. B

4. C

5. C

6. C

7. A

8. C

9. A

10. B

11. D

Page 540

12. B

13. D

14. A

15. C

16. B

17. A

18. B

19. B

20. D

Bonus: B

Chapter 12 Answer KeyForm 1A Form 1B


Chapter 12 Answer Key

Page 541

1. A

2. D

3. C

4. D

5. B

6. D

7. B

8. B

9. C

10. D

11. A

Page 542

12. B

13. A

14. D

15. A

16. C

17. B

18. C

19. A

20. A

Bonus: C

Page 543

1. �23

�

2. �7�54�

3. 537.3

4. �15

2,56625

� or 0.16384

5. 2,222,222.2

6, 6�3�, 18, 18�3�, 54 or 6, �6�3�, 18,

6. �18�3�, 54

7. does not exist

8. 12�2� � 12

9. �43925

�

10. convergent

11. divergent

Page 54427 � 9 � 3 � 1 �

�13

� � �19

�; �1892� or

12. 20�92�

13. �12

k�5 �3(2

2k

k� 1)�

1 � 5�3� � 30 �14. 30�3� � 45 � 9�3�

15. 240x3y8

16. 12.84

17. i� � 2.5416

7 � 3i, 22 � 9i,18. 67 � 27i

�3 � i, 9 � 5i,19. 57 � 89i

20. See students’ work.

Bonus: 8 � 17i

Form 1C Form 2A


Page 545

1. �4

2. �94.4

3. �1440

4. 243

5. 42.5

�4, �12, �36,�108, �324 or �4,

6. 12, �36,108, �324

7. 0

8. 36

9. �9101�

10. convergent

11. divergent

Page 546

12. 2 � 6 � 12 � 20; 40

13.

14.

15. 35,840x3y4

16. 0.7071

17. i� � 2.5953

18.

19.


Bonus: 128

Page 547

1. 3

2. 0.3

3. 1403

4. �2560

5. �1023

6. 9, 3, 1, �13

�

7. 2

8. does not exist

9. �9593�

10. convergent

11. divergent

Page 548

12. 12 � 15 � 18 � 21; 66

13. �8

k�1�k(k

2�k

1)�

14.

15. �216xy3

16. 0.5878

17. i� + 4.0604

18.

19.

20.See students’ work.

Bonus: 32

Chapter 12 Answer KeyForm 2B Form 2C

Sample answer:

�10

n�1�n

n(n

��

11)

�

16p4 � 96p3q �

216p2q2 � 216pq3 �81q4

2 � i, 1 � 0.5i, 0.5 � 0.25i

1 � 2i, �3 � 6i,�27 � 34i

16p4 � 32p3 �24p2 � 8p � 1

2 � 8i, 4 � 16i, 8 � 32i

1 � i, 3i, �9 � i


Chapter 12 Answer KeyCHAPTER 12 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts arithmetic and geometric sequences and series,common differences and ratios of terms, the binomial theorem, and mathematical induction.

• Uses appropriate strategies to solve problems and prove a formula by mathematical induction.

• Computations are correct.• Written explanations are exemplary.• Word problems concerning arithmetic and geometric sequences are appropriate and make sense.

• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts arithmetic and with Minor geometric sequences and series, common differences Flaws and ratios of terms, the binomial theorem, and

mathematical induction.• Uses appropriate strategies to solve problems and prove a formula by mathematical induction.

• Computations are mostly correct.• Written explanations are effective.• Word problems concerning arithmetic and geometric sequences are appropriate and make sense.

• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts arithmetic Satisfactory, and geometric sequences and series, common differences with Serious and ratios of terms, the binomial theorem, and Flaws mathematical induction.

• May not use appropriate strategies to solve problems or prove a formula by mathematical induction.

• Computations are mostly correct.• Written explanations are satisfactory.• Word problems concerning arithmetic and geometric sequences are appropriate and sensible.

• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts arithmetic and geometric sequences and series, common differences and ratios of terms, the binomial theorem, and mathematical induction.

• May not use appropriate strategies to solve problems or prove a formula by mathematical induction.

• Computations are incorrect.• Written explanations are not satisfactory.• Word problems concerning arithmetic and geometric sequences are not appropriate or sensible.

• Does not satisfy requirements of problems.



Page 5491a. Sample answer: Mr. Ling opened a savings

account by depositing $50. He plans todeposit $25 more per month into theaccount. What is his total deposit afterthree months? The sequence is 50 �(n � 1)25, and $100 is his total deposit afterthree months.

1b. Sample answer: The common difference is$25. The nth term is $50 � (n � 1)$25.

1c. Sample answer:

S12 � �122�(50 � 325) � 2250

S12 � 50 � 75 � 100 � 125 � 150 � 175 �

200 � 225 � 250 � 275 � 300 � 325

S12 � (50 � 325) � (75 � 300) �(100 � 275) � (125 � 250) �(150 � 225) � (175 � 200)

Since the sums in parentheses are all equal.

S12 � 6(50 � 325), or �122� (50 � 325), or

�2n� (a1 � an)

1d. No; arithmetic series have no limits; it isdivergent.

2a. Sample answer: Mimi has $60 to spend onvacation. If she spends half of her moneyeach day, how much will she have left afterthe third day?

$60 � ��12

��3

� $7.50

After the third day, she has $7.50.

2b. Sample answer: The common ratio is �12

�.The nth term is 60��1

2��

n�1.

2c. Sample answer:

S11 � � 120

2d. If r � limn→∞

�aan�

n

1� � 1, the series

converges; r � �12

�; the series converges.

3a. Prove that the statement is true for n � 1.Then prove that if the statement is true forn, then it is true for n � 1.

3b. Here Sn is defined as

a1 � a1r � a1r2 � . . . � a1r

n�1 � �a1

1�

�

ar1r

n

�

Step 1: Verify that the formula is valid for n � 1.

Since S1 � a1 and S1 � �a1

1�

�

ar1r

1

�

� �a1

1(1

�

�

rr)

�

� a1,

the formula is valid for n � 1.

Step 2: Assume that the formula is for n � k and derive a formula for n � k � 1.

Sk ⇒ a1 � a1r � a1r2 � . . . � a1r

k�1 �

�a1

1�

�

ar1r

k

�

Sk�1 ⇒ a1 � a1r � a1r2 � . . . � a1r

k�1 �

a1rk�1 � 1

� �a1

1�

�

ar1r

k

� � a1rk�1 � 1

� �a1

1�

�

ar1r

k

� � a1rk

�

�

Apply the original formula for n � k � 1.

Sk�1 ⇒�a1 �

1a�

1rr

( k�1)

�

The formula gives the same result as adding the (k � 1) term directly. thus, if the formula is valid for n � k, it is also validfor n � k � 1. Since the formula is valid forn � 2, it is valid for n � 3, it is also valid forn � 4, and so on indefinitely. Thus, theformula is valid for all integral values of n.

4. From the binomial expansion, the fourth term

of ��y2x��

�yx�

��6is

�36!3!!

��y2x��3��

�yx�

��3� �20 �

y1

3� or ��2y03�.

a1 � a1rk�1

��1 � r

a1 � a1rk � a1r

k � a1rk�1

��1 � r

60 � 60��12

��11

��1 � �

12�

Open-Ended Assessment


Mid-Chapter TestPage 550

1. 129

2. 1175

3. 27 or 128

4. �29,524

5. 64, �32, 16, �8

6. �21�

7. does not exist

8. �6939� or �

171�

9. convergent

10. divergent

Quiz APage 551

1. �9�3� � 9�5�

2. 17

3. 103.7

4. 9�5�

5. 20.78125

6. �13

�, � ��95��, �

257�, . . .

Quiz BPage 551

1. does not exist

2. 1

3. 0

4. �31�

5. does not exist

6. �4959� or �

151�

7. divergent

8. divergent

9. convergent

10. divergent

Quiz CPage 552

1. 2�12

� � 4�12

� � 8�12

�; 15�12

�

2. ��

n�1�1861� ��

23��

n�1

3. �100

k=1(2k � 1)(2k)

81a4 � 108a3d �4. 54a2d2 � 12ad3 � d4

5. 36.77

6. 0.8660

Quiz DPage 552

1. 1.1, 1.11, 1.111, 1.1111

2. 6 � i, 12 � i, 24 � i

3. �2 � 2 i, �1 � 6i, �36 �14i




Sample answer:


Page 553

1. D

2. A

3. C

4. C

5. D

6. E

7. E

8. A

9. C

Page 554

10. D

11. D

12. D

13. E

14. B

15. D

16. B

17. A

18. D

19. 4000

20. 48

Page 555

1. (�5, 7)

No, ƒ(x) is2. undefined when x � 3.

3. (x � 2) (2x � 1) (x � 3)

4. y � �sin �3x � ��5

�� 2

5. �152�92�9��

6. 560.2 N, 32.2

7. 12�cos �96�� i sin �96

��; �12i

15(x�)2 � 30 x�y� �8. 15( y�)2 � 108

9. $1704

10. �10

n�1�(�

21n)n

�

�1

13n

�

Chapter 12 Answer KeySAT/ACT Practice Cumulative Review


Trigonometry Semester Test

Page 557

1. B

2. C

3. B

4. D

5. D

6. B

7. C

8. B

9. A

Page 558

10. C

11. D

12. C

13. D

14. C

15. B

16. C

17. D

18. C

19. C

Page 559

20. A

21. B

22. A

23. B

24. A

25. D

Answer Key


Page 560

26. ��2, 14, 10�

27. y � ��21�x � �1

21�

28. 0.79

29. 1, 3, 9, 27

30. �2�2�, ��4

��

31. (�1, 1)

32. cos 120

33. 0, 30

34. �4�13

1�3��

35. ��47��

36. �5

k�12k

37. 3

38. 4.3938

Page 56139. 3

40. �48,384c5d3

41. �4, 4, 2�

42. �3�13

1�3��

43. �9, 11�

r � csc � or44. 1 � r cos ��

2��

45. x � 0

46. �1�1�

47. yes

48. 11.64

49. �171�

50.

TrigonometryFinal TestPage 563

1. D

2. C

3. C

4. B

5. C

Answer Key


Page 5646. A

7. A

8. D

9. C

10. D

11. B

12. A

13. C

14. B

Page 565

15. A

16. D

17. B

18. C

19. B

20. D

21. D

22. B

23. C

Page 566

24. B

25. B

26. B

27. A

28. C

29. D

30. B

Answer Key


Page 567

31. �4

32. 618.5 cm2

33. � �34. (�1, 3)

35. y � �3x�� 1

36.

37.

38. �1475�

39. � �40. 1

41. 8.1 ft

42. 1541

Page 568

43.

44. 19 yrs

45.

46. 1.3917

47. �1.3673

48. 1,594,322

49. ��6� �4

�2��

50. � �

51. 11

Page 569

52. 2.52 cm

53.

54. log3 �19

� � �2

55. ��6� �4

�2��

56. 0.62

57. 3i� � 4j� � 2k�

58. �9

n�1(2n � 1)

59. �1�3� (cos 5.30 � i sin 5.30)

60. 15 � 8i

61. (�2, �2)

62. �4; imaginary

63. 60°

64. 5; 3

�25

1�2

02

113

27

30

Answer Key

yesD � {�2, 0, 4, 9}R � {�3, 3, 5}

Sample answer: (2, 135°), (�2, �45°),(2, �225°), (�2, 315°)


Page 570

65. y � �13

�x � �23

�

66. (2, 0, �1)

67. 5

68. �21�

69. 36

70. �32�

71. 3 or 1

72. ��13

�, ��32�, �1, �2

73. 28.9 cm

74.

75.

Answer Key

c � 66.4, C � 76.7°,b � 66.3

a � 6.9, B � 107.9°,C � 32.1°

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Chapter 12 Resource Mastersrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812... · 1 5 ( 7) or 2 The common difference is 2. Add 2 to the third term to get the fourth term, and so