# If the sum of first 6 terms is 9 times to the sum of first 3 terms of the same G.P., then common ratio of the series will be

If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value, then the sequence is called a geometric progression. (GP), whereas the constant value is called the common ratio.

The next term of the sequence is produced when we multiply a constant (which is non-zero) to the preceding term. It is represented by:

a, ar, ar2, ar3, ar4, and so on.

#### Sum of GP for Infinite Terms

If the number of terms in a GP is not finite, then the GP is called infinite GP. The formula to find the sum of an infinite geometric progression is

Sn = a(rn-1)/(r-1), where a is the first term and r is the common ratio.

Here,

Sn = Sum of n terms geometric progression

a = First term of G.P.

r = Common ratio of G.P.

n is number of terms

### Solution

Let a be the first term and r be the common ratio of the GP.

Sum of n terms of GP, Sn = a(rn-1)/(r-1)

Given S6 = 9S3

a(r6-1)/(r-1) = 9 a(r3-1)/(r-1)

(r6-1) = 9 (r3-1)

(r3-1)(r3+1) = 9(r3-1)

(r3+1) = 9

r3 = 8

r = 2

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