If the fifth term of a $G.P.$ is $2$,then the product of its first $9$ terms is:

The sum of the series $5.05 + 1.212 + 0.29088 + \dots \infty$ is

The least positive integer $n$ such that $1 - \frac{2}{3} - \frac{2}{3^2} - \dots - \frac{2}{3^{n-1}} < \frac{1}{100}$ is

Consider two $G$.Ps. $2, 2^{2}, 2^{3}, \ldots$ and $4, 4^{2}, 4^{3}, \ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}{8}}$,then $\sum_{k=1}^{n} k(n-k)$ is equal to.

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

Three numbers are in $G.P.$ such that their sum is $38$ and their product is $1728$. The greatest number among them is