10.2

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Analyze Arithmetic Sequences and Series

10.2 Warm-Up

1. 0, 3, 6, 9, 12, …

2. 13, 8, 3, –2, –7, …

ANSWER Each is 3 more than the previous term.

ANSWER Each is 5 less than the previous term.

How is each term in the sequence related to theprevious term?

10.2 Warm-Up

3. 3, 6, 9, 12, …

4. –8, –16, –24, –32, …

ANSWER an = 3n; a5 = 15

ANSWER an = –8n; a5 = –40

Write a rule for the nth term of the sequence. Then find a5.

10.2 Example 1

Tell whether the sequence is arithmetic.a. –4, 1, 6, 11, 16, . . . b. 3, 5, 9, 15, 23, . . .

SOLUTION

Find the differences of consecutive terms.

a2 – a1 = 1 – (–4) = 5a.

a3 – a2 = 6 – 1 = 5

a4 – a3 = 11 – 6 = 5

a5 – a4 = 16 – 11 = 5

b. a2 – a1 = 5 – 3 = 2

a3 – a2 = 9 – 5 = 4

a4 – a3 = 15 – 9 = 6

a5 – a4 = 23 – 15 = 8

10.2

Each difference is 5, so the sequence is arithmetic.

ANSWER ANSWER

The differences are not constant, so the sequence is not arithmetic.

Example 1

10.2 Guided Practice

1. Tell whether the sequence 17, 14, 11, 8, 5, . . . is arithmetic. Explain why or why not.

ANSWER Arithmetic; There is a common differences of –3.

10.2

a. 4, 9, 14, 19, . . . b. 60, 52, 44, 36, . . .

SOLUTION

The sequence is arithmetic with first term a1 = 4 and common difference d = 9 – 4 = 5. So, a rule for the nth term is:an = a1 + (n – 1) d

= 4 + (n – 1)5

= –1 + 5n

Write general rule.

Substitute 4 for a1 and 5 for d.

Simplify.

The 15th term is a15 = –1 + 5(15) = 74.

Write a rule for the nth term of the sequence. Then find a15.

a.

Example 2

10.2

The sequence is arithmetic with first term a1 = 60 and common difference d = 52 – 60 = –8. So, a rule for the nth term is:

an = a1 + (n – 1) d

= 60 + (n – 1)(–8)

= 68 – 8n

Write general rule.

Substitute 60 for a1 and – 8 for d.

Simplify.

b.

The 15th term is a15 = 68 – 8(15) = –52.

Example 2

10.2

One term of an arithmetic sequence is a19 = 48. The common difference is d = 3.

an = a1 + (n – 1)d

a19 = a1 + (19 – 1)d

48 = a1 + 18(3)

Write general rule.

Substitute 19 for n

Solve for a1.

So, a rule for the nth term is:

a. Write a rule for the nth term. b. Graph the sequence.

–6 = a1

Substitute 48 for a19 and 3 for d.

SOLUTION

a. Use the general rule to find the first term.

Example 3

10.2

an = a1 + (n – 1)d

= –6 + (n – 1)3= –9 + 3n

Write general rule.

Substitute –6 for a1 and 3 for d.

Simplify.

Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence.

b.

Example 3

10.2

Two terms of an arithmetic sequence are a8 = 21 and a27 = 97. Find a rule for the nth term.

SOLUTION

STEP 1

Write a system of equations using an = a1 + (n – 1)d and substituting 27 for n (Equation 1) and then 8 for n (Equation 2).

Example 4

10.2

STEP 2 Solve the system. 76 = 19d

4 = d

97 = a1 + 26(4)

Subtract.

Solve for d.

Substitute for d in Equation 1.

–7 = a1 Solve for a1.

STEP 3 Find a rule for an. an = a1 + (n – 1)d Write general rule.

= –7 + (n – 1)4 Substitute for a1 and d.

= –11 + 4n Simplify.

a27 = a1 + (27 – 1)d 97 = a1 + 26da8 = a1 + (8 – 1)d 21 = a1 + 7d

Equation 1

Equation 2

Example 4

10.2

Write a rule for the nth term of the arithmetic sequence. Then find a20.2. 17, 14, 11, 8, . . .

ANSWER an = 20 – 3n; –40

3. a11 = –57, d = –7

ANSWER an = 20 – 7n; –120

4. a7 = 26, a16 = 71

ANSWER an = –9 + 5n; 91

Guided Practice

10.2

SOLUTION

a1 = 3 + 5(1) = 8

a20 = 3 + 5(20) =103

S20 = 20 ( )8 + 103 2

= 1110

Identify first term.

Identify last term.

Write rule for S20, substituting 8 for a1 and 103 for a20.Simplify.

ANSWER The correct answer is C.

Example 5

10.2

You are making a house of cards similar to the one shown.

Write a rule for the number of cards in the nth row if the top row is row 1.

a.

What is the total number of cards if the house of cards has 14 rows?

b.

House Of Cards

Example 6

10.2

SOLUTION

Starting with the top row, the numbers of cards in the rows are 3, 6, 9, 12, . . . . These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is:

a.

an = a1 + (n – 1) = d

= 3 + (n – 1)3= 3n

Write general rule.

Substitute 3 for a1 and 3 for d.

Simplify.

Example 6

10.2

SOLUTION

Total number of cards = S14 = 14( )a1 + a14

2= 14( )3 + 42

2= 315

Find the sum of an arithmetic series with first term a1 = 3 and last term a14 = 3(14) = 42.

b.

Example 6

10.2

5. Find the sum of the arithmetic series (2 + 7i).

12

i = 1ANSWER S12 = 570

6. WHAT IF? In Example 6, what is the total number of cards if the house of cards has 8 rows?

ANSWER 108 cards

Guided Practice

10.2

1.

Is the sequence 2, 103, 204, 305, 406, . . . arithmetic? Explain your answer.

Write a rule for the nth term of the sequence 5, 2, –1, –4, . . .. Then find a5.

2.

Two terms of an arithmetic sequence are a5 = 14 and a30 = 89. Find a rule for the nth term.

3.

ANSWER Yes; the common difference is 101.

ANSWER an = 8 – 3n ; a5 = –7

ANSWER an = –1 + 3n

Lesson Quiz

10.2

5. How many seats are in the theater?

A movie theater has 24 seats in the first row and each successive row contains one additional seat. There are 30 rows in all.

ANSWER 1155 seats

4.

Write a rule for the number of seats in the nth row.

ANSWER an = 23 + n

Lesson Quiz