The ratio of the sum of first three terms is to that of first six terms of a GP
Question:

The ratio of the sum of first three terms is to that of first six terms of a GP is 125 : 152. Find the common ratio.

Solution:

The first three terms of a G.P. are:a,ar,ar2

The first six terms of a G.P. are:a,ar, $a r^{2}, a r^{3}, a r^{4}, a r^{5}$

It is given that the ratio of the sum of first three terms is to that of first six terms of a GP is $125: 152$.

$\Rightarrow a+a r+a r^{2}=125 x \& a+a r+a r^{2}+a r^{3}+a r^{4}+a r^{5}=152 x$

$\Rightarrow a+a r+a r^{2}+r^{3}\left(a+a r+a r^{2}\right)=152 x$

$\Rightarrow 125 x+r^{3}(125 x)=152 x$

$\Rightarrow r^{3}(125 x)=152 x-125 x=27 x$

$\Rightarrow r^{3}=\frac{27}{125}=\left(\frac{3}{5}\right)^{3}$

⇒ r = 3/5

Ans: common ratio $=\frac{3}{5}$