# The sum of the first 6 terms of a geometric sequence

The sum of the first 6 terms of a geometric sequence is 9 times the sum of its first 3 terms. Find the common ratio.

A)    3
B)    4
C)    2
D)    1
E)    7

Solution

Sum of n terms of a geometric series is given by a ( r^n – 1) / r – 1    where a, r and n are the first term, ratio and number of terms of the series respectively.

Plugging in values, we get,

=> (a (r ^ 6 – 1) / r – 1) / (a (r ^ 3 – 1) / r – 1) = 9
=> (r ^ 6 – 1) / (r ^ 3 – 1) = 9

Assuming r ^ 3 as x, we get

=>  (x ^ 2 – 1) / (x – 1) = 9
=>  (x ^ 2 – 1) = 9 (x – 1)
=>  x ^ 2 – 9x + 8  = 0
=>  x = 8

Thus r ^ 3 = 8, and r = 2

Common ratio of the series is 2

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