1. Arithmetic Sequences

2. A mathematical model for the average annual salaries of major league baseball players generates the following data. 1,438,0001,347,0001,256,0001,165,0001,074,000983,000892,000801,000 Salary 19981997199619951994199319921991 Year From 1991 to 1992, salaries increased by $892,000 - $801,000 = $91,000. From 1992 to 1993, salaries increased by $983,000 - $892,000 = $91,000. If we make these computations for each year, we find that the yearly salary increase is $91,000. The sequence of annual salaries shows that each term after the first, 801,000, differs from the preceding term by a constant amount, namely 91,000. The sequence of annual salaries 801,000, 892,000, 983,000, 1,074,000, 1,165,000, 1,256,000.... is an example of an arithmetic sequence. Arithmetic Sequences

3. Definition of an Arithmetic Sequence An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence.

4. The recursion formula a n  a n  1  24 models the thousands of Air Force personnel on active duty for each year starting with 1986. In 1986, there were 624 thousand personnel on active duty. Find the first five terms of the arithmetic sequence in which a 1  624 and a n  a n  1  24. Solution The recursion formula a n  a n  1  24 indicates that each term after the first is obtained by adding  24 to the previous term. Thus, each year there are 24 thousand fewer personnel on active duty in the Air Force than in the previous year. Text Example

5. The recursion formula a n  a n  1  24 models the thousands of Air Force personnel on active duty for each year starting with 1986. In 1986, there were 624 thousand personnel on active duty. Find the first five terms of the arithmetic sequence in which a 1  624 and a n  a n  1  24. The first five terms are 624, 600, 576, 552, and 528. Solution a 1  624 This is given. a 2  a 1 – 24  624 – 24  600 Use a n  a n  1  24 with n  2. a 3  a 2 – 24  600 – 24  576 Use a n  a n  1  24 with n  3. a 4  a 3 – 24  576 – 24  552 Use a n  a n  1  24 with n  4. a 5  a 4 – 24  552 – 24  528 Use a n  a n  1  24 with n  5. Text Example cont.

6. Example Write the first six terms of the arithmetic sequence where a 1 = 50 and d = 22 a 1 = 50 a 2 = 72 a 3 = 94 a 4 = 116 a 5 = 138 a 6 = 160 Solution:

7. General Term of an Arithmetic Sequence The nth term (the general term) of an arithmetic sequence with first term a 1 and common difference d is a n = a 1 + (n-1)d

8. Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is  7. Solution To find the eighth term, as, we replace n in the formula with 8, a 1 with 4, and d with  7. a n  a 1  (n  1)d a 8  (8  1)(  7)  4  )  4  (  49)   5 The eighth term is  45. We can check this result by writing the first eight terms of the sequence: 4,  3,  10,  17,  24,  31,  38,  45. Text Example

9. The Sum of the First n Terms of an Arithmetic Sequence The sum, S n, of the first n terms of an arithmetic sequence is given by in which a 1 is the first term and a n is the nth term.

10. Example Solution: Find the sum of the first 20 terms of the arithmetic sequence: 6, 9, 12, 15,...

11. Example Find the indicated sum Solution:

12. Arithmetic Sequences