Question
What is the product of first $$2n + 1$$ terms of a geometric progression ?
A.
The $${\left( {n + 1} \right)^{th}}$$ power of the $$n^{th}$$ term of the G.P.
B.
The $${\left( {2n + 1} \right)^{th}}$$ power of the $$n^{th}$$ term of the G.P.
C.
The $${\left( {2n + 1} \right)^{th}}$$ power of the $${\left( {n + 1} \right)^{th}}$$ term of the G.P.
D.
The $$n^{th}$$ power of the $${\left( {n + 1} \right)^{th}}$$ term of the G.P.
Answer :
The $${\left( {2n + 1} \right)^{th}}$$ power of the $${\left( {n + 1} \right)^{th}}$$ term of the G.P.
Solution :
The G.P. is $$a,ar,a{r^2},.....,a{r^{2n}}$$
So, $$P = a \cdot ar \cdot a{r^2} \cdot ..... \cdot a{r^{2n}}$$
$$\eqalign{
& = {a^{2n + 1}} \cdot {r^{1 + 2 + ..... + 2n}} \cr
& = {a^{\left( {2n + 1} \right)}}{r^{\frac{{2n\left( {2n + 1} \right)}}{2}}} = {a^{2n + 1}}{r^{n\left( {2n + 1} \right)}} = {\left( {a{r^n}} \right)^{\left( {2n + 1} \right)}} \cr} $$
$$ = {\left( {2n + 1} \right)^{th}}$$ power of the $${\left( {n + 1} \right)^{th}}$$ term of G.P.