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 2016

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 t years. 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. Find the nth term of the following sequence: The nth Term of a Geometric Sequence 15, 75, 375, 1875,... The nth term is 15(5) n-1.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Find the 9th term of the 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.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Definition of Sequence Find the sum of the first 8 terms of the series: The sum of the terms of a geometric sequence is called a geometric series.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 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. 13 The Sum of a Finite Geometric Sequence Example: Find the 8 th partial sum of the geometric series: …

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Evaluate the summation:

Homework Day 1: 8.3 Pg , odd odd Day 2: 8.3 Pg odd +87 Quiz on sections 8.1 – 8.3 on Wednesday Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

Day 2 Geometric Series 2016

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 HWQ For the geometric sequence where the 3 rd term is 5/3 and the 6 th term is 5/81, find the general expression.

Warm-Up Your twin sister lands a great job that pays her a salary of $20,000 per month. She laughs at your new job that pays $.01 the first day, $.02 the second day, $.04 the third day, and so on. Find your total income after working 31 days if your daily wage keeps doubling. Who’s laughing now? $21,474, Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Definition of Sequence Write the series using summation notation. The sum of the terms of an infinite geometric sequence is called an infinite geometric series.

Evaluate the summation using the finite geometric sum formula : Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Graphing Utility: Terms and Sum of a Sequence Graphing Utility: Find the sum List Menu:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Example: Evaluate the following summation.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Example: Evaluate the following summation. Evaluate the sum by using (and simplifying) the finite geometric sum formula.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Definition of Geometric Series The sum of an infinite geometric series If |r| < 1, then an infinite geometric series and the series is said to converge to that sum. has the sum In this case, the series is said to diverge.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Example: Sum of Infinite Geometric Series In the series The sum of the series is

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Example: Sum of Infinite Geometric Series Example: Find the sum of

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Example: Sum of Infinite Geometric Series Example: Find the sum of

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 Example: Find the rational representation for:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Example: Find the rational representation for:

Bouncing ball problem Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 A ball is dropped from a height of 6 ft. and begins bouncing. The height of each bounce is three fourths the height of the previous bounce. Find the total vertical distance traveled by the ball.

You are so square Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31 This is a square of side length 4 inches. The midpoints of each side are the vertices of an inscribed square. The pattern repeats forever. Show that the perimeters of the squares form a geometric sequence. Does the related series converge? If so, find its sum.

You are so square Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33 Definition of Sequence When will a geometric series converge? Make sure you know this:

Homework 8.3 Day 2: Pg odd, +87 Quiz on sections 8.1 – 8.3 Tomorrow Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 34

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35 Definition of Sequence Find the number of terms needed for the sum of the series to exceed 2,000,000. Just for fun:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 36 $2500 is invested at a rate of 4%, compounded monthly. How much will the investment be worth in 20 years?