Show that the products of the corresponding terms of the sequences a, ar, ar2,…arn – 1 and A, AR, AR2, … ARn – 1 form a G.P, and find the common ratio.
Let the two given sequences be
\[ a,\; ar,\; ar^2,\; \ldots,\; ar^{n-1} \]
and
\[ A,\; AR,\; AR^2,\; \ldots,\; AR^{n-1}. \]
We are required to show that the products of the corresponding terms of these two sequences form a geometric progression and to find its common ratio.
The products of the corresponding terms are:
\[ a \cdot A,\; ar \cdot AR,\; ar^2 \cdot AR^2,\; \ldots,\; ar^{n-1} \cdot AR^{n-1}. \]
Simplifying each term, we get
\[ aA,\; aA(rR),\; aA(rR)^2,\; \ldots,\; aA(rR)^{n-1}. \]
Clearly, the above sequence has first term
\[ aA \]
and each term is obtained by multiplying the previous term by
\[ rR. \]
Therefore, the products of the corresponding terms form a geometric progression.
The common ratio of this G.P. is
\[ \boxed{rR}. \]
If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P2 = (ab) n .