Geometric Sequences and Series Section 9-3

2 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

3 Definition of a Geometric Sequence A 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.

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

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.

6 General Term of a Geometric Sequence The nth term (the general term) of a geometric sequence with the first term a 1 and common ratio r is a n = a 1 r n-1

7 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: a n = 15(5) n-1. a 2 = 15(5) a 3 = 15(5 2 ) a 4 = 15(5 3 )

8 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

9 The Sum of the First n Terms of a Geometric Sequence The sum, S n, of the first n terms of a geometric sequence is given by in which a 1 is the first term and r is the common ratio.

10 Example Find the sum of the first 12 terms of the geometric sequence: 4, -12, 36, -108,... Solution:

11 The sum of the first n terms of a sequence is represented by summation notation. Definition of Summation Notation index of summation upper limit of summation lower limit of summation

12 The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by = ? n = 8 a 1 = 5

13 The Sum of an Infinite Geometric Series If -1<r<1, then the sum of the infinite geometric series a 1 +a 1 r+a 1 r 2 +a 1 r 3 +… in which a 1 is the first term and r s the common ration is given by If |r|>1, the infinite series does not have a sum.

14 Example: Sum of Infinite Geometric Series Example: Find the sum of The sum of the series is

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17 Homework WS 13-5