1. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 Definition of Sequence You find a job that pays an annual salary of $32,000 with an annual raise of $2500. How much money would you make in the first 10 years of employment? Warm-Up

2. Geometric Sequences and Series 2015

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 A deposit of $5000 is made in an account that earns 3% interest compounded quarterly. a)Find an equation that expresses the balance in the account after n quarters. b)Find the balance in the account after 10 years. c)Is this an arithmetic sequence?

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 An infinite sequence is a function whose domain is the set of positive integers. a 1, a 2, a 3, a 4,..., a n,... Definition of Sequence The first three terms of the sequence a n = 2n 2 are a 1 = 2(1) 2 = 2 a 2 = 2(2) 2 = 8 a 3 = 2(3) 2 = 18. finite sequence terms

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512,... Definition of Geometric Sequence geometric sequence The common ratio, r, is 4.

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 The nth term of a geometric sequence has the form a n = a 1 r n - 1 where r is the common ratio of consecutive terms of the sequence. The nth Term of a Geometric Sequence 15, 75, 375, 1875,... a 1 = 15 The nth term is 15(5) n-1. a 2 = 15(5) a 3 = 15(5 2 ) a 4 = 15(5 3 )

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Find the 9th term of the geometric sequence 7, 21, 63,... Example: Finding the nth Term a 1 = 7 The 9th term is 45,927. a n = a 1 r n – 1 = 7(3) n – 1 a 9 = 7(3) 9 – 1 = 7(3) 8 = 7(6561) = 45,927

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Write a rule for the sequence, then find the 10 th term.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 For the geometric sequence where the 4 th term is 125 and the 10 th term is 125/64, find the general expression and then find the 14 th term. (Assume that all terms of the sequence are positive )

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 You Try: For the geometric sequence where the 5 th term is 1/8 and the 8 th term is -1/64, find the general expression and then find the 12 th term.

11. Homework Day 1: Pg. 582 1-41 odd Day 2: Pg. 582 45-77 odd Quiz on sections 8.1 – 8.3 on Friday Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

12. Day 2 Geometric Series 2015

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 The Sum of a Finite Geometric Sequence Example: Find the sum of the finite geometric sequence: 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a 1 = 5

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Find the following sum:

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Definition of Geometric Series The sum of the terms of an infinite geometric sequence is called a geometric series. a 1 + a 1 r + a 1 r 2 + a 1 r 3 +... + a 1 r n-1 +... If |r| < 1, then the infinite geometric series has the sum

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Example: Sum of Infinite Geometric Series Example: Find the sum of The sum of the series is

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Example: Sum of Infinite Geometric Series Example: Find the sum of

19. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Example: Sum of Infinite Geometric Series Example: Find the sum of

20. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Graphing Utility: Terms and Sum of a Sequence Graphing Utility: Find the first 5 terms of the geometric sequence a n = 2(1.3) n. List Menu: variable Graphing Utility: Find the sum List Menu: beginning value end value variable upper limit lower limit

21. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Find the rational representation for:

22. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 You Try: Find the rational representation for:

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 $2500 is invested at a rate of 4%, compounded monthly. How much will the investment be worth in 20 years?

24. Homework Day 1: Pg. 582 1-41 odd Day 2: Pg. 582 45-77 odd Quiz on sections 8.1 – 8.3 on Friday Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24

25. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Definition of Sequence Write the series using summation notation, then find the sum. Pre-Quiz Warm-up

26. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Definition of Sequence Write the sum using summation notation. Pre-Quiz Warm-up

27. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Definition of Sequence Find the sum of the first 20 terms. Pre-Quiz Warm-up

28. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 Definition of Sequence Find the sum of the first 50 terms. Then find the sum of an infinite number of terms. Pre-Quiz Warm-up

29. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Definition of Sequence Simplify. Pre-Quiz Warm-up