For a $G$.$P$.,if $S_n = \frac{4^n - 3^n}{3^n}$,then $t_2 = ........$

If $1 + r + r^2 + \dots + r^n = (1 + r) (1 + r^2) (1 + r^4) (1 + r^8)$,then what is the value of $n$?

If the $p^{\text{th}}$,$q^{\text{th}}$,and $r^{\text{th}}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,prove that $a^{q-r} b^{r-p} c^{p-q} = 1$.

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?

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 $A \cdot H = \dots$

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$,then $24(a_1 + a_2 + a_3)$ is equal to