Geometric Sequences and Series Digital Lesson Geometric Sequences and Series

Definition of Sequence An infinite sequence is a function whose domain is the set of positive integers. a1, a2, a3, a4, . . . , an, . . . terms The first three terms of the sequence an = 2n2 are a1 = 2(1)2 = 2 a2 = 2(2)2 = 8 finite sequence a3 = 2(3)2 = 18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition of Sequence

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

The nth Term of a Geometric Sequence The nth term of a geometric sequence has the form an = a1rn - 1 where r is the common ratio of consecutive terms of the sequence. a1 = 15 15, 75, 375, 1875, . . . a2 = 15(5) a3 = 15(52) a4 = 15(53) The nth term is 15(5n-1). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The nth Term of a Geometric Sequence

Example: Finding the nth Term Example: Find the 9th term of the geometric sequence 7, 21, 63, . . . a1 = 7 an = a1rn – 1 = 7(3)n – 1 a9 = 7(3)9 – 1 = 7(3)8 = 7(6561) = 45,927 The 9th term is 45,927. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Finding the nth Term

Definition of Summation Notation The sum of the first n terms of a sequence is represented by summation notation. upper limit of summation lower limit of summation index of summation Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition of Summation Notation

The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a1 = 5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The Sum of a Finite Geometric Sequence

Definition of Geometric Series The sum of the terms of an infinite geometric sequence is called a geometric series. If |r| < 1, then the infinite geometric series a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . . has the sum Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition of Geometric Series

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. Example: Sum of Infinite Geometric Series

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