arithmetic sequences and partial sums 8.2
- 1. Copyright © Cengage Learning. All rights reserved. 8.2 Arithmetic Sequences and Partial Sums
- 2. 2 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.
- 4. 4 Arithmetic Sequences A sequence whose consecutive terms have a common difference is called an arithmetic sequence.
- 5. 5 Example 1 – Examples of Arithmetic Sequences a. The sequence whose nth term is 4n + 3 is arithmetic. The common difference between consecutive terms is 4. 7, 11, 15, 19, . . . , 4n + 3, . . . Begin with n = 1. 11 – 7 = 4
- 6. 6 Example 1 – Examples of Arithmetic Sequences b. The sequence whose nth term is 7 – 5n is arithmetic. The common difference between consecutive terms is –5. 2, –3, –8, –13, . . . , 7 – 5n, . . . Begin with n = 1. –3 – 2 = –5 cont’d
- 7. 7 Example 1 – Examples of Arithmetic Sequences c. The sequence whose nth term is is arithmetic. The common difference between consecutive terms is Begin with n = 1. cont’d
- 9. 9 The Sum of a Finite Arithmetic Sequence
- 10. 10 The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence.
- 11. 11 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
- 12. 12 Example 5 – Solution = 5(20) = 100. cont’d Substitute 10 for n, 1 for a1, and 19 for an. Simplify.
- 13. 13 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.
- 14. 14 Applications
- 15. 15 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 a1 = 20000 and d = 15,000. So, an = 20,000 + 15,000(n – 1)
- 16. 16 Example 7 – Solution and the nth term of the sequence is an = 15,000n + 5000. This implies that the 20th term of the sequence is a20 = 15,000(20) + 5000 = 300,000 + 5000 = 305,000. cont’d
- 17. 17 Example 7 – Solution The sum of the first 20 terms of the sequence is = 10(325,000) = 3,250,000. So, the total sales for the first 20 years are $3,250,000. cont’d Simplify. Substitute 20 for n, 20,000 for a1, and 305,000 for an. nth partial sum formula Simplify.