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 :
$\frac{5}{2}$
$\frac{1}{2}$
$\frac{25}{2}$
$\frac{9}{2}$
The sum of the first five terms of the series $3 + 4\frac{1}{2} + 6\frac{3}{4} + ......$ will be
If the sum of first 6 term is $9$ times to the sum of first $3$ terms of the same $G.P.$, then the common ratio of the series will be
The first term of an infinite geometric progression is $x$ and its sum is $5$. Then
The sum of first three terms of a $G.P.$ is $16$ and the sum of the next three terms is
$128.$ Determine the first term, the common ratio and the sum to $n$ terms of the $G.P.$
If $x$ is added to each of numbers $3, 9, 21$ so that the resulting numbers may be in $G.P.$, then the value of $x$ will be