Question

The ratio of the sum of first $3$ terms to that of the first $6$ terms of a $G.P.$ is $125:152$.Find their common ratio's.

Answer 1

$S_n=\frac{a{r}^n-1}{r-1}$

$=\frac{S_3}{S_6}=\frac{a(r^3-1)/r-1}{a(r^6-1)/r-1}=\frac{125}{152}$

$=\frac{r^3-1}{r^6-1}=\frac{125}{152}$

$=152r^3-152=125r^6-125$

$=125r^6-152r^3-27=0$

$=125r^3(r^3-1)-27(r^3-1)=0$

$=(125r^3-27)(r^3-1)=0$

$r^3=\frac{27}{125}$ $r^3=1$

$r=\frac{3}{5}$ $r=1$

$\therefore r=\frac{3}{5}$.