Question

Find the sum to $n$ terms of a G.P., $1,-a,a^2,-a^3....$ (if $a\ne -1$)

Answer 1

Let $A$ be the first term and $R$ be the common ratio.
$\therefore$ first term $=A=1$
and common ratio $=R=\frac{-a}{1}=-a$
We know, the sum of $n$ terms of G.P. is
$S_n=\frac{A(1-R^n)}{1-R}$
$\Rightarrow S_n=\frac{1(1-(-a{)}^n)}{1-(-a)}$
$\therefore S_n=\frac{1-(-a{)}^n}{1+a}$