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{ar^{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{ar^{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 $r^{n}$.