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

Geometric Sequences and Series 2015

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?

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

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.

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 )

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

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

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 )

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.

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

Day 2 Geometric Series 2015

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

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: = ? n = 8 a 1 = 5

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

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 a 1 r n If |r| < 1, then the infinite geometric series has the sum

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

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

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

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

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

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

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?

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

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

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

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

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

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