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 a n+1 = a n + d for any integer n  1.

Example For each of the following arithmetic sequences, identify the first term, a 1, and the common difference, d. a)6, 10, 14, 18, 22, … b)0,  6,  12,  18,  24, … c) Solution: The first term a 1 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.

Example continued We obtained the common difference by subtracting a 1 from a 2. Had we subtracted a 2 from a 3, or a 3 from a 4, 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, a 1 c)  6 (  6  0 =  6) 0 b)0,  6,  12,  18,  24, … 4 (10  6 = 4) 6a)6, 10, 14, 18, 22, … Common Difference, dSequence

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: a n = a 1 + (n  1)d, for any integer

Example Find the 11 th term of the arithmetic sequence 2, 6, 10, 14, … Solution: We first note that a 1 = 2, d = 4, and n = 11. Then using the formula for the nth term, we obtain a n = a 1 + (n  1)d a 11 = 2 + (11  1)4 a 11 = a 11 = 42 The 11 th term is 42.

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

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

Example Find the sum of the first 11 terms of the arithmetic sequence 16, 12, 8, 4, … Solution: Note that a 1 = 16, d =  4, and n = 11. First we find the last term a 11. a 11 = 16 + (11  1)(  4) = 16  40 =  24 Thus, The sum of the first 11 terms is  44.

Example Find the sum:. Solution: It is helpful to write out a few terms first:   . It appears that a 1 = 14, d = 10, n = 10. We then find the last term. a n = a 1 + (n – 1)d a 10 = 14 + (10 – 1)10 = 104 Thus,

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 a 8 to find the number of musicians in the last row. a 8 = 5 + (8  1)2 a 8 = 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.