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

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Solution

First sequence is $a, a r, a r^{2}, a r^{3}, . . ., a r^{n - 1}$

Second sequence is $A, A R, A R^{2}, A R^{3}, . . ., A R^{n - 1}$

Let $P =$ product of corresponding terms of the first and second sequence.

$= a A, a r A R, a r^{2} A R^{2}, a r^{3} A R^{3}, . . ., a r^{n - 1} A R^{n - 1}$

Now, $\frac{a r A R}{a A} = r R$ and $\frac{a r^{2} A R^{2}}{a r A R} = r R$

Thus, the common ratio is the same.

thus, it is a G.P.

$∴$ Common ratio $= r R$

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