Question:
Find the sum to n terms in the geometric progression $1,-a, a^{2},-a^{3} \ldots($ if $a \neq-1)$
Solution:
The given G.P. is $1,-a, a^{2},-a^{3}, \ldots \ldots \ldots \ldots . .$
Here, first term $=a_{1}=1$
Common ratio $=r=-a$
$\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{a}_{1}\left(1-\mathrm{r}^{\mathrm{n}}\right)}{1-\mathrm{r}}$
$\therefore S_{n}=\frac{1\left[1-(-a)^{n}\right]}{1-(-a)}=\frac{\left[1-(-a)^{n}\right]}{1+a}$