Question

Show that the sum of $n$ terms of a $G.P$ common ratio $r$ beginning with the $p^{th}$ term is $r^{p-q}$ times the sum of an equal number of terms of the same series beginning with $q^{th}$ term.

Answer 1

Let $S_1$ be sum of $n$ terms of G.P starting with $p^{th}$ term and $S_2$ be for $q^{th}$ term then,

$S_1=a{r}^{p-1}\frac{(r^n-1)}{r-1}$

$S_2=a{r}^{q-1}\frac{(r^n-1)}{(r-1)}$

$\therefore S_1=r^{p-q}{S}_2$

Thus proved.