Question

The sum of the first $6$ terms of a G.P. is $9$ times the sum of the first $3$ terms; find the common ratio.

Answer 1

Let $S_n$ be sum first $n$ terms of G.P. and $r$ be common ratio
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$
Given :
$S_6=9\times S_3$

$S_6=\frac{a(r^6-1)}{r-1}=9\times \frac{a(r^3-1)}{r-1}$
$\Rightarrow r^6-1=9(r^3-1)$

$\Rightarrow (r^3-1)(r^3+1)=9(r^3-1)$
$\Rightarrow r^3=-1+9=8$
$\Rightarrow r=2$