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10.3 Geometric Sequences 10.3 Geometric Sequences and Series If we start with a number, a1, and repeatedly multiply it by some constant, r, then we have a geometric sequence: a1, a1r, a1r 2, a1r 3, a1r 4,…. The nth term of a geometric sequence is given n 1 by an a1r , for any integer n 1 The number r is called the common ratio 10.3 Geometric Sequences and Series (Example 2) Find the 8th term of the geometric sequence: 5, 15, 45, … Solution: Use formula, an = a1r (n – 1) a1 = r= n= (Now work Example 3 in text….) Geometric Series The sum of the terms of a geometric sequence is called a geometric series. For example: S n a ar ar 2 ar 3 .... ar n 2 ar n 1 is a finite geometric series with common ratio r. What is the sum of the n terms of a finite geometric series? Deriving a formula for the nth partial sum of a geometric series S n a ar ar 2 .... ar n 2 ar n 1 rS n ar ar 2 + .... + ar n 2 ar n 1 ar n S n rS n a 0 0 . . . . . 0 0 ar n S n (1 r ) a(1 r n ) (1 r n ) Sn a (1 r ) 10.3 Geometric Sequences and Series The sum of the terms of an infinite geometric sequence is an infinite geometric series. For some geometric sequences, Sn gets close to a specific number as n gets large. This number becomes the limit of the sum of the infinite geometric sequence. When |r|<1, the limit or sum of an infinite geometric series is given by S a1 . 1 r 10.3 Geometric Sequences and Series You should be able to: Identify the common ratio of a geometric sequence, and find a given term and the sum of the first n terms. Find the sum of an infinite geometric series, if it exists. 10.1 Sequences and Series You should be able to: Find terms of sequences given the nth term. Look for a pattern in a sequence and try to determine a general term. Convert between sigma notation and other notation for a series. Construct the terms of a recursively defined sequence.

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