Question

In a G.P the ratio of the sum of the first $3$ terms is to that of first $6$ terms is $125:152$. find the common ratio

Answer 1

Formula,

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

Given,

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

$\frac{r^3-1}{(r^3-1)(r^3+1)}=\frac{125}{152}$

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

$152=125r^3+125$

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

$\therefore$ Common ratio $r=\frac{3}{5}$