Let $a_1, a_2, ..., a_{10}$ be a $G.P.$ If $\frac{a_3}{a_1} = 25$,then $\frac{a_9}{a_5}$ is equal to:

The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

Let $a_{n}$ be the $n^{\text{th}}$ term of a $G$.$P$. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$,then $\sum_{n=1}^{200} a_{n}$ is equal to:

How many terms of the $G.P.$ $3, 3^{2}, 3^{3}, \dots$ are needed to give the sum $120$?

If the $(m + n)^{th}$ term of a geometric progression is $9$ and the $(m - n)^{th}$ term is $4$,what is the $m^{th}$ term?

Let $G_1$ and $G_2$ be the geometric means of two series $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ respectively. If $G$ is the geometric mean of the series $\frac{x_i}{y_i}$ where $i = 1, 2, \dots, n$,then what is $G$ equal to?