1. 7 Further Topics in Algebra © 2008 Pearson Addison-Wesley. All rights reserved Sections 7.1–7.3

2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-2 7.1 Sequences and Series 7.2 Arithmetic Sequences and Series 7.3 Geometric Sequences and Series Further Topics in Algebra 7

3. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-3 Sequences and Series 7.1 Sequences ▪ Series and Summation Notation ▪ Summation Properties

4. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-4 Write the first five terms for each sequence. 7.1 Example 1 Finding Terms of Sequences (page 653)

5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-5 7.1 Example 1 Finding Terms of Sequences (cont.)

6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-6 Write the first four terms for each sequence. 7.1 Example 2 Using a Recursion Formula (page 654)

7. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-7 Suppose an insect population density in thousands per acre during year n can be modeled by the recursively defined sequence 7.1 Example 3(a) Modeling Insect Population Growth (page 655) Find the population for n = 1, 2, 3, 4, 5, 6.

8. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-8 7.1 Example 3(a) Modeling Insect Population Growth (cont.)

9. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-9 7.1 Example 3(a) Modeling Insect Population Growth (cont.) The figure shows the computation of the sequence, denoted by u(n) rather than a n using a calculator.

10. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-10 7.1 Example 3(b) Modeling Insect Population Growth (page 655) Describe what happens to the population density as n increases. As n increases, the population density oscillates above and below 6.4 thousand insects per acre. It may stabilize near 6.4 thousand per acre.

11. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-11 Evaluate the series 7.1 Example 4 Using Summation Notation (page 657) Write each of the five terms, then evaluate the sum.

12. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-12 Write the terms for each series. Evaluate each sum, if possible. 7.1 Example 5 Using Summation Notation with Subscripts (page 657)

13. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-13 7.1 Example 5 Using Summation Notation with Subscripts (cont.)

14. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-14 7.1 Example 6 Using the Summation Properties (page 658) Use the summation properties to find each sum.

15. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-15 7.1 Example 6 Using the Summation Properties (cont.)

16. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-16 7.1 Example 7 Using the Summation Properties (page 659) Summation rules

17. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-17 Arithmetic Sequences and Series 7.2 Arithmetic Sequences ▪ Arithmetic Series

18. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-18 Find the common difference, d, for the arithmetic sequence 20, 13, 6, –1, –8, … 7.2 Example 1 Finding the Common Difference (page 664)

19. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-19 Find the first five terms for each arithmetic sequence. 7.2 Example 2 Finding Terms Given a 1 and d (page 664) (a)The first term is –14, and the common difference is 6. Starting with a 1 = –14, add d = 6 to each term to get the next term.

20. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-20 7.2 Example 2 Finding Terms Given a 1 and d (cont.) (b)a 1 = 1.5, d = –.5 Starting with a 1 = 1.5, add d = –.5 to each term to get the next term.

21. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-21 7.2 Example 3 Finding Terms of an Arithmetic Sequence (page 665) Find a 16 and a n for the arithmetic sequence 23, 20, 17, 14, …. so

22. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-22 7.2 Example 4 Finding Terms of an Arithmetic Sequence (page 665) Find a 10 and a n for the arithmetic sequence having a 4 = 18 and a 5 = 22. so

23. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-23 7.2 Example 5 Finding the First Term of an Arithmetic Sequence (page 666) Suppose that an arithmetic sequence has a 6 = 9 and a 15 = –36. Find a 1. First find d: Use the equation a 6 = a 1 + 5d to find a 1.

24. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-24 7.2 Example 6 Finding the n th Term From a Graph (page 666) Find a formula for the nth term of the sequence a n shown in the graph. What are the domain and range of this sequence?

25. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-25 7.2 Example 6 Finding the n th Term From a Graph (cont.) The sequence is comprised of the points {(1, 1), (2, 3), (3, 5), (4, 7), (5, 9)}. The points lie on a line with slope Using the point-slope form of a line, we have

26. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-26 7.2 Example 6 Finding the n th Term From a Graph (cont.) The nth term of the sequence is determined by

27. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-27 7.2 Example 7 Using the Sum Formulas (page 668) (a)Evaluate S 21 for the arithmetic sequence 48, 44, 40, 36, …. so

28. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-28 7.2 Example 7 Using the Sum Formulas (cont.) (b)Use a formula for S n to evaluate the sum of the first 200 positive integers. The first 200 positive integers form the sequence 1, 2, 3, 4, …, 200. Thus, n = 200, a 1 = 1, and a 200 = 200. so

29. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-29 7.2 Example 8 Using the Sum Formulas (page 669) The sum of the first 15 terms of an arithmetic sequence is 345. If a 15 = 65, find a 1 and d. Sum formula S 15 = 345, a 15 = 65 Formula for nth term of an arithmetic sequence. a 15 = 65, a 1 = –19

30. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-30 7.2 Example 9 Using Summation Notation (page 669) Evaluate each sum. The first few terms of the series are This is the sum of an arithmetic sequence with d = 5 and a 1 = 15. If i started at 1, there would be 12 terms. Since i starts at 5, there are four terms missing, and n = 8.

31. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-31 7.2 Example 9 Using Summation Notation (cont.) Using the formula we have

32. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-32 7.2 Example 9 Using Summation Notation (cont.) This is the sum of an arithmetic sequence with d = –2 and a 1 = –2. If k started at 1, there would be 10 terms. Since k starts at 5, there are four terms missing, and n = 6. The first few terms of the series are

33. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-33 7.2 Example 9 Using Summation Notation (cont.) Using the formula we have

34. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-34 Geometric Sequences and Series 7.3 Geometric Sequences ▪ Infinite Geometric Series ▪ Annuities

35. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-35 Suppose that you receive a gift on the first day of each month for a year, starting with $50 on January 1, with the amount doubling each month. How much will you receive on December 1? 7.3 Example 1 Finding the n th Term of a Geometric Sequence (page 673) The first term, a 1, is 50 and r = 2 since the amount doubles each month. You will receive $102,400 on December 1.

36. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-36 7.3 Example 2 Finding Terms of a Geometric Sequence (page 673) Find a 5 and a n for the geometric sequence 6400, 1600, 400, 100, …. First find r:

37. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-37 7.3 Example 3 Finding Terms of a Geometric Sequence (page 673) Find r and a 1 for the geometric sequence with second term –18 and fifth term 486. Use the formula for the nth term of a geometric sequence. (1) (2) Substitute the value of r from equation (1) into equation (2).

38. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-38 7.3 Example 3 Finding Terms of a Geometric Sequence (cont.) Now find r :

39. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-39 7.3 Example 4 Modeling a Population of Fruit Flies (page 674) A population of fruit flies is growing in such a way that each generation is 1.75 times as large as the last generation. Suppose there are 250 insects in the first generation. How many would there be in the sixth generation? a 1 = 250 and r = 1.75 There will be about 4103 insects in the sixth generation.

40. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-40 7.3 Example 5 Finding the Sum of the First n Terms (page 675) Suppose that you receive a gift on the first day of each month for a year, starting with $50 on January 1, with the amount doubling each month. What is the total amount of the gifts you will receive throughout the year? a 1 = 50 and r = 2 Using the summation formula where r ≠ 1, we have You will receive a total of $204,750.

41. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-41 7.3 Example 6 Finding the Sum of the First n Terms (page 676) Find This series is the sum of the first 8 terms of a geometric sequence having a 1 = 4 ∙ 5 1 = 20 and r = 5. Using the summation formula where r ≠ 1, we have

42. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-42 7.3 Example 7 Summing the Terms of an Infinite Geometric Series (page 676) and Use the formula for the sum of the first n terms of a geometric sequence to obtain In general,

43. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-43 7.3 Example 7 Summing the Terms of an Infinite Geometric Series (cont.) As n gets larger, approaches 0, or Thus,

44. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-44 7.3 Example 8 Finding the Sum of the Terms of an Infinite Geometric Series (page 678) Find each sum. The series converges because –1 < r < 1, so

45. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-45 7.3 Example 8 Finding the Sum of the Terms of an Infinite Geometric Series (cont.) The series converges because –1 < r < 1, so

46. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-46 7.3 Example 9 Finding the Future Value of an Annuity (page 678) To save money for a new car, Heather deposited $3500 at the end of each year for 6 years in an account paying 5% interest, compounded annually. Find the future value of this annuity. The future value is the sum of the payments. Using the formula for compound interest, the first year payment is 3500(1.05) 6, the second year payment is 3500(1.05) 5, etc., with the fifth year payment 3500(1.05) 1, and the last year payment 3500(1.05) 0.

47. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-47 7.3 Example 9 Finding the Future Value of an Annuity (cont.) This is the sum of a geometric sequence with a 1 = 3500, r = 1.05, and n = 6. The future value of the annuity is about $23,806.69.