ss_11_3

ss_11_3 - Section 11.3 Geometric sequences A geometric...

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A geometric sequence { a n } is a sequence that can be written in the following form. a 1 = a a n = ra n - 1 where r 6 = 0. The number r is called the common ratio . Example. { 1 , 3 , 9 , 27 , ···} is a geometric sequence with common ratio r =3. Example. { 1 , - 1 , 1 , - 1 , ···} is a geometric sequence with common ratio r = - 1. Example. Is the sequence { a n } = { 2 n } a geometric sequence? Solution. a n +1 a n = 2 n +1 2 n = 2. Since a n +1 a n is a constant, the sequence is a geometric sequence with r =2. Example. Is the sequence { n +1 n } = { a n } a geometric sequence? Solution. a n +1 a n = n +2 n +1 n +1 n = n ( n +2) ( n +1) 2 . Since a n +1 a n is not constant, the sequence is not a geometric sequence. Finding the n-th term of a geometric sequence: a 1 = a a 2 = ar a 3 = ar 2 ··· a n = ar n - 1 The above shows that the n -th term, a n , of a geometric sequence is a n = ar n - 1 Sum of a geometric sequence: Let S n be the sum of the first n terms of a geometric sequence. Then S 1 = a 1 S 2 = a + ar = a (1 + r ) S 3 = a + ar + ar 2 = a (1 + r + r 2 ) ··· S n = a + ar + ar 2 + ··· + ar n - 1 = a (1 + r + r 2
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ss_11_3 - Section 11.3 Geometric sequences A geometric...

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