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