A geometric series has first term a and common ratio r . The second term of the series is 4 and the sum up to infinite terms of the series is 25. If r is taken as the larger of its two possible values

Question

A geometric series has first term a and common ratio r . The second term of the series is 4 and the sum up to infinite terms of the series is 25. If r is taken as the larger of its two possible values, then the smallest value of n for which S n ​ exceeds 15 is

TestHub · Accepted Answer

The larger value is r = 5 4 ​ with a = 5 . S n ​ = 1 − 5 4 ​ 5 ( 1 − ( 5 4 ​ ) n ) ​ = 25 ( 1 − ( 5 4 ​ ) n ) > 15 , so ( 5 4 ​ ) n < 5 2 ​ . This is 2 2 n − 1 < 5 n − 1 . For n = 4 : 2 7 < 5 3 is false. For n = 5 : 2 9 < 5 4 is true. Smallest n = 5 .

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