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

(N/A) The given sequences are $a, ar, ar^{2}, \dots, ar^{n-1}$ and $A, AR, AR^{2}, \dots, AR^{n-1}$.
Their corresponding products are $aA, (ar)(AR), (ar^{2})(AR^{2}), \dots, (ar^{n-1})(AR^{n-1})$,which simplifies to $aA, (arR), (arR)^{2}, \dots, (arR)^{n-1}$.
To check if this is a $G.P.$,we find the ratio of consecutive terms:
$\frac{\text{Second term}}{\text{First term}} = \frac{arAR}{aA} = rR$
$\frac{\text{Third term}}{\text{Second term}} = \frac{ar^{2}AR^{2}}{arAR} = rR$
Since the ratio of consecutive terms is constant,the sequence forms a $G.P.$ with a common ratio of $rR$.