Section 11.2

Arithmetic Sequences and

Series

Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives

• For any arithmetic sequence, find the nth term when n is given and n when the nth term is given, and

given two terms, find the common difference and construct the sequence.• Find the sum of the first n terms of an arithmetic sequence.

Arithmetic Sequences

A sequence in which each term after the first is found by adding the same number to the preceding term is an arithmetic sequence.

A sequence is arithmetic if there exists a number d, called the common difference, such that an+1 = an + d for any integer n 1.

Example

For each of the following arithmetic sequences, identify the first term, a1, and the common difference, d.a) 6, 10, 14, 18, 22, …b) 0, 6, 12, 18, 24, …

c)

Solution: The first term a1 is the first term listed. To find the common difference, d, we choose any term beyond the first and subtract the preceding term from it.

1 2 4 5, ,1, , ,...3 3 3 3

Example continued

We obtained the common difference by subtracting a1 from a2. Had we subtracted a2 from a3, or a3 from a4, we would have obtained the same values for d. We can check by adding d to each term in a sequence to see if we progress correctly to the next term.

First Term, a1

13

c)

6 (6 0 = 6)0b) 0, 6, 12, 18,24, …

4 (10 6 = 4)6a) 6, 10, 14, 18, 22, …

Common Difference, dSequence

1 2 4 5, ,1, , ,...3 3 3 3

1 2 1 1 3 3 3 3

13

nth Term of an Arithmetic Sequence

To find a formula for the general, or nth, term of any arithmetic sequence, we denote the common difference by d, write out the first few terms, and look for a pattern.

The nth term of an arithmetic sequence is given by the formula: an = a1 + (n 1)d, for any integer 1n

Example

Find the 11th term of the arithmetic sequence 2, 6, 10, 14, …

Solution: We first note that a1 = 2, d = 4, and n = 11. Then using the formula for the nth term, we obtain an = a1 + (n 1)d a11 = 2 + (11 1)4 a11 = 2 + 40 a11 = 42The 11th term is 42.

Example

The 3rd term of an arithmetic sequence is 5, and the 9th term is 37. Find a1 and d and construct the sequence. Solution We know that a3 = 5 and a9 = 37. Thus we have to add d 6 times to get to 37 from 5. 5 + 6d = 37 6d = 42 d = 7 Since a3 = 5, we subtract d twice to get a1. a1= 5 2(7) = 19 The sequence is 19, 12, 5, 2, …In general, d should be subtracted n 1 times from an in order to find a1.

Sum of the First n Terms

The formula for the sum of the first n terms of an arithmetic sequence is given by:

( ).2n 1 nnS = a +a

Example

Find the sum of the first 11 terms of the arithmetic sequence 16, 12, 8, 4, …

Solution: Note that a1 = 16, d = 4, and n = 11. First we find the last term a11. a11 = 16 + (11 1)(4) = 16 40 = 24

Thus,

The sum of the first 11 terms is 44.

1111 11(16 24) ( 8) 44.2 2

S

Example

Find the sum: .

Solution: It is helpful to write out a few terms first: 14 + 24 + 34 + .It appears that a1 = 14, d = 10, n = 10. We then find the last term. an = a1+ (n – 1)d

a10 = 14 + (10 – 1)10 = 104 Thus,

10

1

(10 4)k

k

1010 (14 104)2

5(118)590.

S

Example

An orchestra consists of 8 rows of musicians. The first row has 5 musicians, the second row has 7 musicians, and the third row has 9 musicians.a) How many musicians are in the last row?b) What is the total number of musicians in the orchestra?

Solution: a) We need to find a8 to find the number of musicians in

the last row. a8 = 5 + (8 1)2a8 = 19

There are 19 musicians in the last row.

Example continued

b) We can then use the formula to find the total number of musicians.

There are a total of 96 musicians in the orchestra.

( )2n 1 nnS = a +a

1

8

8

8

( )28 (5 19)24(24)96

n nnS a a

S

SS