Pre-Algebra 12-2 Geometric Sequences Check 12-1 HOMEWORK

Pre-Algebra

12-2 Geometric Sequences

Check 12-1 HOMEWORK

Pre-Algebra


Pre-Algebra HOMEWORK

Page 606

#10-18

Pre-Algebra


Students will be able to solve sequences and represent functions by completing the following assignments.

• Learn to find terms in an arithmetic sequence.• Learn to find terms in a geometric sequence.• Learn to find patterns in sequences.• Learn to represent functions with tables, graphs, or equations.

Pre-Algebra


Today’s Learning Goal Assignment

Learn to find terms in a geometric sequence.

Pre-Algebra

12-2 Geometric Sequences12-2 Geometric Sequences

Pre-Algebra

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra


Warm Up1. Determine if the sequence could be arithmetic. If so, give the common difference. 100, 50, 25, 12.5, . . .

Find the given term in each arithmetic sequence.

2. 12th term: a1 = 30, d = 0.5

3. 55th term: 4, 28, 52, 76

no

35.5

1300

Pre-Algebra


Pre-Algebra


Problem of the Day

Two students begin counting by 3’s at the same time. One counts up from 0, and the other counts down from 120. If each says one number every second, will both students ever say the same number at the same time?yes (60)

Pre-Algebra


Vocabulary

geometric sequencecommon ratio

Pre-Algebra


In a geometric sequence, the ratio of one term to the next is always the same. This ratio is called the common ratio. The common ratio is multiplied by each term to get the next term.

Pre-Algebra


Determine if the sequence could be geometric. If so, give the common ratio.

A. 1, 5, 25, 125, 625, …

Additional Example 1A: Identifying Geometric Sequences

The sequence could be a geometric with a common ratio of 5.

Divide each term by the term before it.

1 5 25 125 625, . . .

5555

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A. 2, 10, 50, 250, 1250, . . .

Try This: Example 1A



2 10 50 250 1250, . . .

5555

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B. 1, 3, 9, 12, 15, …

Additional Example 1B: Identifying Geometric Sequences

The sequence is not geometric.


1 3 9 12 15, . . .

54

43

33

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B. 1, 1, 1, 1, 1, . . .

Try This: Example 1B



1 1 1 1 1, . . .

1111

Pre-Algebra



C. 81, 27, 9, 3, 1, . . .

Additional Example 1C: Identifying Geometric Sequences

The sequence could be geometric with a common ratio of .1

3


81 27 9 3 1, . . .

13

13

13

13

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C. 2, 4, 12, 24, 96, . . .

Try This: Example 1C

The sequence is not geometric.


2 4 12 24 96, . . .

4232

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D. –3, 6, –12, 24, –48

Additional Example 1D: Identifying Geometric Sequences

The sequence could be geometric with a common ratio of –2.


–3 6 –12 24 –48, . . .

–2–2–2–2

Pre-Algebra



D. 1, 2, 4, 8, 16, . . .

Try This: Example 1D

The sequence could be geometric with a common ratio of 2.

1 2 4 8 16, . . .

2222


Pre-Algebra


FINDING THE nth TERM OF A GEOMETRIC SEQUENCE

The nth term an of a geometric sequence with common ratio r is

an = a1rn–1.

Pre-Algebra


Find the given term in the geometric sequence.

A. 11th term: –2, 4, –8, 16, . . .

Additional Example 2A: Finding a Given Term of a Geometric Sequence

an = a1rn–1

a11 = –2(–2)10 = –2(1024) = –2048

r = = –2 4–2

Pre-Algebra



A. 12th term: -2, 4, -8, 16, . . .

Try This: Example 2A

an = a1rn–1

a12 = –2(–2)11 = –2(–2048) = 4096

r = = –2 4–2

Pre-Algebra



B. 9th term: 100, 70, 49, 34.3, . . .

Additional Example 2B: Finding a Given Term of a Geometric Sequence

an = a1rn–1

a9 = 100(0.7)8 = 100(0.05764801) = 5.764801

r = = 0.7 70 100

Pre-Algebra



B. 11th term: 100, 70, 49, 34.3, . . .

Try This: Example 2B

a11 = 100(0.7)10 = 100(0.0282475249) 2.825

an = a1rn–1

r = = 0.7 70100

Pre-Algebra



C. 10th term: 0.01, 0.1, 1, 10, . . .

Additional Example 2C: Finding a Given Term of a Geometric Sequence

an = a1rn–1

a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000

r = = 10 0.10.01

Pre-Algebra



C. 5th term: 0.01, 0.1, 1, 10, . . .

Try This: Example 2C

a5 = 0.01(10)4 = 0.01(10,000) = 100

an = a1rn–1

r = = 10 0.10.01

Pre-Algebra



D. 7th term: 1000, 200, 40, 8, . . .

Additional Example 2D: Finding a Given Term of a Geometric Sequence

an = a1rn–1

r = = 2001000

15

a7 = 1000( )6 = 1000( )= , or 0.064

15

8 125

1 15,625

Pre-Algebra



D. 12th term: 1000, 200, 40, 8, …

Try This: Example 2D

an = a1rn–1

r = = 2001000

15

a5 = 1000 ( )4 = 1000( )= , or 1.6

15

85

1 625

Pre-Algebra


Tara sells computers. She has the option of earning (1) $50 per sale or (2) $1 for the first sale, $2 for the second sale, $4 for the third sale and so on, where each sale is worth twice as much as the previous sale. If Tara estimates that she can sell 10 computers a week, which option should she choose?

Additional Example 3: Money Application

If Tara chooses $50 per sale, she will get a total of 10($50) = $500.

Pre-Algebra


Additional Example 3 Continued

a10 = ($1)(2)9 = ($1)(512) = $512

Option 1 gives Tara more money in the beginning, but option 2 gives her a larger total amount.

If Tara chooses the second option, her earnings for just the 10th sale will be more that the total of all the earnings in option 1.

Pre-Algebra


A gumball machine at the mall has 932 gumballs. If 19 gumballs are bought each day, how many gumballs will be left in the machine on the 7th day?

Try This: Example 3

a7 = (932)(0.98)6 (932)(0.89) 829

an = a1rn–1

r = 0.98913932

n = 7 a1 = 932

There will be about 829 gumballs in the machine after 7 days.

Pre-Algebra


Lesson QuizDetermine if each sequence could be geometric. If so, give the common ratio.

1. 200, 100, 50, 25, 12.5, . . .

2. 4, 8, 12, 16, . . .

Find the given term in each geometric

sequence.

3. 7th term: , 1, 3, 9, . . .

4. 20th term: a1 = 800, r = 0.8

no

243

≈ 11.53

13

yes; 12

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Pre-Algebra 12-2 Geometric Sequences Check 12-1 HOMEWORK