Show that the products of the corresponding terms of the sequences $a, a r, {a r}^{2}, {a r}^{n - 1}, a n d A, A R, {A R}^{2}, {A R}^{n - 1}$ from a G.P. and find the common ratio.

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Solution

It has to be proved that the sequence , $a A, a r A R, {a r}^{2} {A R}^{2}, . . . {a r}^{n - 1} {A R}^{n - 1},$ forms a G.P.
$\frac{Second Term}{First Term} = \frac{a r A R}{a A} = r R$
$\frac{Third Term}{Second Term} = \frac{{a r}^{2} {A R}^{2}}{a r A R} = r R$
Thus, the above sequence forms a G.P. with common ratio $r R .$

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