Question

A G.P consists of $2n$ terms. If the sum of the terms occupying the odd place is $S_1$ and that of occupying the even places is $S_2$ then find the common ratio of the progression

Answer 1

Given,

$S_1=a+a{r}^2+...+a{r}^{2n-2}$

$\Rightarrow S_1=a(1+r^2+......+r^{2n-2})$

$S_2=ar+a{r}^3+.....+a{r}^{2n-1}$

$\Rightarrow S_2=ar(1+r^2+......+r^{2n-2})$

$\frac{S_2}{S_1}=\frac{ar(1+r^2+......+r^{2n-2})}{a(1+r^2+......+r^{2n-2})}$

$=r$

$\therefore r=\frac{S_2}{S_1}$