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

What is the geometric mean of $3, 3^2, 3^3, \dots, 3^n$?

The product of three consecutive terms of a geometric progression is $216$ and the sum of the products of these terms taken two at a time is $156$. Find the terms.

Let $a$ and $b$ be roots of $x^2 - 3x + p = 0$ and let $c$ and $d$ be the roots of $x^2 - 12x + q = 0$,where $a, b, c, d$ form an increasing $G$.$P$. Then the ratio of $(q + p) : (q - p)$ is equal to

If $|\alpha| < 1$ and $|\beta| < 1$,and $1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty = s_1$ and $1 - \beta + \beta^2 - \beta^3 + \dots \infty = s_2$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals:

Consider a geometric progression with first term $a$ and common ratio $r$. If $A$ and $H$ are the arithmetic mean and harmonic mean of the first $n$ terms of the geometric progression respectively,then $A \cdot H = \dots$