Question

If $s$ represents the sum of $n$ terms of G.P whose first term and common ratio are $a$ and $r$ respectively, then $s_1+s_2+s_3+...+s_n$

Answer 1

As we know when $r>1$,

Summation in $GP$-

$s_n=\frac{a(r^n-1)}{r-1}$, where $a$ is first term.

Now,

Let $S=s_1+s_2+s_3+.......+s_n$

$S=\frac{a(r^1-1)}{r-1}+\frac{a(r^2-1)}{r-1}+\frac{a(r^3-1)}{r-1}+.....+\frac{a(r^n-1)}{r-1}$

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

$=\frac{1}{r-1}[\frac{a(r^n-1)}{r-1}-an]$

Or

$S=\frac{a(r^n-1)}{(r-1{)}^2}-\frac{an}{r-1}$

This is the answer.