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**Geometric Sequences and Series**

Digital Lesson Geometric Sequences and Series

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**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

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**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

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**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, , . . . 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

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**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

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**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

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**The Sum of a Finite Geometric Sequence**

The sum of a finite geometric sequence is given by = ? n = 8 a1 = 5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The Sum of a Finite Geometric Sequence

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**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 + a1r a1rn has the sum Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition of Geometric Series

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**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

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**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

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