Question:
Show that the products of the corresponding terms of the sequences $a, a r, a r^{2}, \ldots a r^{n-1}$ and $A, A R, A R^{2}, \ldots A R^{n-1}$ form a G.P, and find the common ratio.
Solution:
It has to be proved that the sequence, $a A, \operatorname{arAR}, a r^{2} A R^{2}, \ldots a r^{n-1} A R^{n-1}$, forms a G.P.
$\frac{\text { Second term }}{\text { First term }}=\frac{\text { ar } A R}{a A}=r R$
$\frac{\text { Third term }}{\text { Second term }}=\frac{a r^{2} A R^{2}}{a r A R}=r R$
Thus, the above sequence forms a G.P. and the common ratio is rR.
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