Show that the products of the corresponding, terms of the sequences a, ar, ar², ….ar^{n – 1} and A, AR, AR²,……AR^{n – 1} form a G.P. and find the common ratio.

2^nd G.P. : A, AR, AR²,….. AR^{n – 1}

The sequence formed after multiplying the corresponding terms of the sequences is

(aA), (aA),(rR), (aR)r²R²,…..(aA)r^{n – 1} R^{n – 1}

Here = = rR

= = rR

= = rR

Since the ratios of two succeeding terms are the same, the resulting sequence is also a G.P.

The common ratio of the new G.P. = (rR).