If the sum of an infinite $GP$ $a, ar, ar^{2}, a r^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150 ,$ then the sum of $\mathrm{ar}^{2}, \mathrm{ar}^{4}, \mathrm{ar}^{6}, \ldots$ is :
$\frac{5}{2}$
$\frac{1}{2}$
$\frac{25}{2}$
$\frac{9}{2}$
If ${x_r} = \cos (\pi /{3^r}) - i\sin (\pi /{3^r}),$ (where $i = \sqrt{-1}),$ then value of $x_1.x_2.x_3......\infty ,$ is :-
The G.M. of the numbers $3,\,{3^2},\,{3^3},\,......,\,{3^n}$ is
Fifth term of a $G.P.$ is $2$, then the product of its $9$ terms is
Let ${a_n}$ be the ${n^{th}}$ term of the G.P. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $, such that $\alpha \ne \beta $,then the common ratio is
The first term of an infinite geometric progression is $x$ and its sum is $5$. Then