The sum of a $G.P.$ with common ratio $r = 3$ is $364$,and the last term is $243$. Find the number of terms $n$.

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 $G_1$ and $G_2$ are two geometric means between two numbers and $A$ is the arithmetic mean between them,then find the value of $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1}$.

If the first term of an infinite geometric series is $1$ and each term is equal to the sum of all the terms that follow it,then what is its fourth term?

If $a_1, a_2, a_3, \ldots, a_{10}$ is a geometric progression and $\frac{a_3}{a_1}=25$,then $\frac{a_9}{a_5}$ equals

If $x, y, z$ are in $G.P.$ and $a^x = b^y = c^z$,then