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# Show that the product of the corresponding terms of the sequences $a,ar,ar^2.........ar^n$ and $A,AR,AR^2.............AR^{n-1}$ form a G.P. and find the common ratio.

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
$aA\bigg(1.\large\frac{1-(rR)^n}{1-rR}\bigg)$
edited Mar 1, 2014