In a G.P., the ratio between the sum of first three terms and that of the first six terms is 125 : 152. Find its common ratio.

Given,

$\Rightarrow \dfrac{S$3}{S6} = \dfrac{125}{152} \\[1em] \Rightarrow \dfrac{\dfrac{a(r^3 - 1)}{r - 1}}{\dfrac{a(r^6 - 1)}{r - 1}} = \dfrac{125}{152} \\[1em] \Rightarrow \dfrac{r^3 - 1}{r^6 - 1} = \dfrac{125}{152} \\[1em] \Rightarrow \dfrac{r^3 - 1}{(r^3 - 1)(r^3 + 1)} = \dfrac{125}{152} \\[1em] \Rightarrow \dfrac{1}{r^3 + 1} = \dfrac{125}{152} \\[1em] \Rightarrow r^3 + 1 = \dfrac{152}{125} \\[1em] \Rightarrow r^3 = \dfrac{152}{125} - 1 \\[1em] \Rightarrow r^3 = \dfrac{152 - 125}{125} \\[1em] \Rightarrow r^3 = \dfrac{27}{125} \\[1em] \Rightarrow r^3 = \Big(\dfrac{3}{5}\Big)^3 \\[1em] \Rightarrow r = \dfrac{3}{5}.

Hence, common ratio = $\dfrac{3}{5}$.