11_3 - S n = a 1 1-r n 1-r if r 6 = 1. 4 Geometric series A...

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Section 11.3: Geometric Sequences Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu) 1 Geometric Sequences A geometric sequence { a n } satisfies the following: a 1 6 = 0 a n = a n - 1 r where a 1 is the first term, r 6 = 0 is the common ratio. So, r = a n a n - 1 . 2 Find the n -th term of a geometric sequence a 1 given, a 2 = a 1 r a 3 = a 2 r = a 1 r 2 ... a n = a n - 1 r = a 1 r n - 1 So, the n -th term of a geometric sequence is: a n = a 1 r n - 1 . Example 1 : Example 2 : 1
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Example 3 : 3 Summing the terms of geometric sequence Let S k ( k = 1 , ..., n ) be the sum of the first k terms of a geometric sequence { a n } . Then S 1 = a 1 S 2 = a 1 + a 2 = a 1 + ( a 1 r ) = a 1 (1 + r ) S 3 = a 1 + a 2 + a 3 = a 1 + ( a 1 r ) + ( a 1 r 2 ) = a 1 (1 + r + r 2 ) ... S n = a 1 + a 2 + a 3 + ... + a n = a 1 + ( a 1 r ) + ( a 1 r 2 ) + ... + [ a 1 r n - 1 ] = a 1 (1 + r + r 2 + ... + r n - 1 ) = a 1 1 - r n 1 - r if r 6 = 1. So, the sum of the first n terms of a geometric sequence is
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Unformatted text preview: S n = a 1 1-r n 1-r if r 6 = 1. 4 Geometric series A geometric series is a series of the form: a 1 + a 1 r + a 1 r 2 + ... = X k =0 a 1 r k = a 1 X k =0 r k In other words, the geometric series is a sum of an innite number of terms of a geometric sequence. If | r | < 1, then S = a 1 + a 1 r + a 1 r 2 + ... = a 1 1 1-r . Example 4 : Example 5 : 2 5 Write Repeating Decimal as a Fraction So, a 1 = 89 100 , r = 1 100 , S = 89 99 . Example 6 : Example 7 : 3...
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11_3 - S n = a 1 1-r n 1-r if r 6 = 1. 4 Geometric series A...

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