Given two sequences are

$a,ar,ar^2.........ar^n$ ...........(i) and

$A,AR,AR^2.............AR^{n-1}$ ..........(ii)

These two sequences are G.P.

The sum of the products of the corresponding terms of (i) and (ii) is

$S_n=a.A+ar.AR+........ar^{n-1}.AR^{n-1}$

Taking $aA$ common

$\Rightarrow\:aA(1+rR+r^2.R^2+.....r^{n-1}.R^{n-1})$

This is a G.P. since the ratio of any two successive terms is same.

$i.e.,$ the common ratio =$\large\frac{rR}{1}=\frac{r^2R^2}{rR}$$=rR$

and the sum of this G,P. is

$aA\bigg(1.\large\frac{1-(rR)^n}{1-rR}\bigg)$