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

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Solution

$G P_{1}$
$\Rightarrow a, a r, a r^{2} . . . \dots \dots \dots . a r^{n - 1}$

$G P_{2}$
$\Rightarrow A, A R, A R^{2} . . . \dots \dots . . A R^{n - 1}$

$G P_{1} \times G P_{2}$
$\Rightarrow a A, a A R r, a A r^{2} R^{2} . . . \dots \dots \dots . . . a A r^{n - 1} R^{n - 1}$

Ratio $\Rightarrow \frac{a A r R}{a A} = r R$ & $\frac{a A r^{2} R^{2}}{a A R r} = r R$

$∴$ Common ratio $= r R$ & it is same through the series

$∴$ It is also a G.P.

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