8 2 Arithmetic Sequences and Partial Sums Copyright

8. 2 Arithmetic Sequences and Partial Sums Copyright © Cengage Learning. All rights reserved.

What You Should Learn • Recognize, write, and find the nth terms of arithmetic sequences. • Find nth partial sums of arithmetic sequences. Make sure you write down this formula on slide #8 and at least one example. • Use arithmetic sequences to model and solve real-life problems. Make sure you write down this formula on slide #10 and at least one example. 2

Arithmetic Sequences 3

Arithmetic Sequences A sequence whose consecutive terms have a common difference is called an arithmetic sequence. 4

Example 1 – Examples of Arithmetic Sequences a. The sequence whose nth term is 4 n + 3 is arithmetic. The common difference between consecutive terms is 4. 7, 11, 15, 19, . . . , 4 n + 3, . . . Begin with n = 1. 11 – 7 = 4 5

Example 1 – Examples of Arithmetic Sequences cont’d b. The sequence whose nth term is 7 – 5 n is arithmetic. The common difference between consecutive terms is – 5. 2, – 3, – 8, – 13, . . . , 7 – 5 n, . . . Begin with n = 1. – 3 – 2 = – 5 6

Example 1 – Examples of Arithmetic Sequences cont’d c. The sequence whose nth term is is arithmetic. The common difference between consecutive terms is Begin with n = 1. 7

Arithmetic Sequences 8

The Sum of a Finite Arithmetic Sequence 9

The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence. 10

Example 5 – Finding the Sum of a Finite Arithmetic Sequence Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19. Solution: To begin, notice that the sequence is arithmetic (with a common difference of 2). Moreover, the sequence has 10 terms. So, the sum of the sequence is Sn = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 Formula for sum of an arithmetic sequence 11

Example 5 – Solution cont’d Substitute 10 for n, 1 for a 1, and 19 for an. = 5(20) Simplify. = 100. 12

The Sum of a Finite Arithmetic Sequence The sum of the first n terms of an infinite sequence is called the nth partial sum. The nth partial sum of an arithmetic sequence can be found by using the formula for the sum of a finite arithmetic sequence which is on slide #10. 13

Applications 14

Example 7 – Total Sales A small business sells $20, 000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $15, 000 each year for 19 years. Assuming that this goal is met, find the total sales during the first 20 years this business is in operation. Solution: The annual sales form an arithmetic sequence in which a 1 = 20000 and d = 15, 000. So, an = 20, 000 + 15, 000(n – 1) 15

Example 7 – Solution cont’d and the nth term of the sequence is an = 15, 000 n + 5000. This implies that the 20 th term of the sequence is a 20 = 15, 000(20) + 5000 = 300, 000 + 5000 = 305, 000. 16

Example 7 – Solution cont’d The sum of the first 20 terms of the sequence is nth partial sum formula Substitute 20 for n, 20, 000 for a 1, and 305, 000 for an. = 10(325, 000) Simplify. = 3, 250, 000. Simplify. So, the total sales for the first 20 years are $3, 250, 000. 17