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Geometric Sequences and Series Section 9-3

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Section 9-3. Geometric Sequences and Series. Objectives. Recognize, write, and find nth terms of geometric sequences Find the nth partial sums of geometric sequences Find the sum of an infinite geometric sequence. Definition of a Geometric Sequence. - PowerPoint PPT Presentation

Geometric Sequences and SeriesSection 9-3

Copyright by Houghton Mifflin Company, Inc. All rights reserved.

*ObjectivesRecognize, write, and find nth terms of geometric sequencesFind the nth partial sums of geometric sequencesFind the sum of an infinite geometric sequence

*Definition of a Geometric SequenceA geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

*An infinite sequence is a function whose domain is the set of positive integers. a1, a2, a3, a4, . . . , an, . . .Definition of SequenceThe first three terms of the sequence an = 2n2 area1 = 2(1)2 = 2a2 = 2(2)2 = 8a3 = 2(3)2 = 18.finite sequenceterms

*A sequence is geometric if the ratios of consecutive terms are the same.2, 8, 32, 128, 512, . . . Definition of Geometric Sequencegeometric sequence The common ratio, r, is 4.

*General Term of a Geometric SequenceThe nth term (the general term) of a geometric sequence with the first term a1 and common ratio r is an = a1 r n-1

*The nth term of a geometric sequence has the form an = a1rn - 1where r is the common ratio of consecutive terms of the sequence.The nth Term of a Geometric Sequence15, 75, 375, 1875, . . . a1 = 15 The nth term is: an = 15(5)n-1.

*Example: Find the 9th term of the geometric sequence7, 21, 63, . . .Example: Finding the nth Terma1 = 7 The 9th term is 45,927.an = a1rn 1 = 7(3)n 1a9 = 7(3)9 1 = 7(3)8= 7(6561) = 45,927

*The Sum of the First n Terms of a Geometric SequenceThe sum, Sn, of the first n terms of a geometric sequence is given by

in which a1 is the first term and r is the common ratio.

*ExampleFind the sum of the first 12 terms of the geometric sequence: 4, -12, 36, -108, ...Solution:

*The sum of the first n terms of a sequence is represented by summation notation. Definition of Summation Notationindex of summationupper limit of summationlower limit of summation

*The Sum of a Finite Geometric SequenceThe sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?n = 8a1 = 5

- *The Sum of an Infinite Geometric SeriesIf -1
*Example: Sum of Infinite Geometric SeriesExample: Find the sum of

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*HomeworkWS 13-5

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