Question

If $S_r$ denotes the sum of the first r terms of a G.P:, then $S_n,S_{2n}-S_n,S_{3n}-S_{2n}$ - are in

• A. A.P
• B. G.P.
• C. H.P
• D. none of these

Answer 1

Answer: (B)

$S_n=a\left(\frac{1-r^n}{1-r}\right);S_{2n}=\frac{a\left(1-r^{2n}\right)}{1-r}$,

$S_{3n}=\frac{a\left(1-r^{3n}\right)}{1-r}$

$S_{2n}-S_n=\frac{a}{1-r}\left[\left(1-r^{2n}\right)-\left(1-r^n\right)\right]$

$=\frac{a\left(r^n-r^{2n}\right)}{1-r}=\frac{a{r}^n\left(1-r^n\right)}{1-r}$

$S_{3n}-S_{2n}=\frac{a\left(1-r^{3n}\right)}{1-r}-\frac{a\left(1-r^{2n}\right)}{1-r}$

$=\frac{a\left(r^{2n}-r^{3n}\right)}{1-r}-\frac{a{r}^{2n}}{1-r}\left[1-r^n\right]$

Clearly $S_n,S_{2n},-S_n,S_{3n}-S_{2n}$ form a $G.P$. with common ratio $r^n$.