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

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It has to be proved that the sequence: $a A, a r A R, a r^{2} A R^{2}, \ldots \ldots a r^{n-1} A R^{n-1},$ forms a $G.P.$

$\frac{{{\rm{ Second}}\,\,{\rm{term }}}}{{{\rm{ First }}\,\,{\rm{term }}}} = \frac{{ar\,AR}}{{a\,A}} = rR$

$\frac{{{\rm{ Third}}\,\,{\rm{ tem }}}}{{{\rm{ Second }}\,\,{\rm{term }}}} = \frac{{a{r^2}\,A{R^2}}}{{ar\,AR}} = rR$

Thus, the above sequence forms a $G.P.$ and the common ratio is $rR.$

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