If the first term of a Geometric Progression $(GP)$ is $1$ and the sum of its third and fifth terms is $90$,find the common ratio.

If the $n^{th}$ term of the geometric progression $5, - \frac{5}{2}, \frac{5}{4}, - \frac{5}{8}, \dots$ is $\frac{5}{1024}$,then the value of $n$ is:

Let $f(x)$ be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1)=3$ and $\sum_{k=1}^{n} f(k)=3279$,then the value of $n$ is:

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?

Let $A_n = \left( \frac{3}{4} \right) - \left( \frac{3}{4} \right)^2 + \left( \frac{3}{4} \right)^3 - \dots + (-1)^{n-1} \left( \frac{3}{4} \right)^n$ and $B_n = 1 - A_n$. Then,the least odd natural number $p$ such that $B_n > A_n$ for all $n \geq p$ is:

The first three terms of a geometric progression are $a, b, c$. If the harmonic mean of $a$ and $b$ is $12$ and the harmonic mean of $b$ and $c$ is $36$,then $a = \dots$