If the first term of an infinite geometric series is twice the sum of all its subsequent terms,what is the common ratio?

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a geometric progression are $a, b, c$ respectively,then $a^{q-r} \cdot b^{r-p} \cdot c^{p-q} = \dots\dots$

Let $C_0$ be a circle of radius $1$. For $n \geq 1$,let $C_n$ be a circle whose area equals the area of a square inscribed in $C_{n-1}$. Then,$\sum_{i=0}^{\infty} \text{Area}(C_i)$ equals

If $n$ geometric means between $a$ and $b$ are $G_1, G_2, ..., G_n$ and a single geometric mean is $G$,then the true relation is

If $x > 1, y > 1, z > 1$ are in geometric progression,then in which progression are $\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}$?

Insert three numbers between $1$ and $256$ so that the resulting sequence is a $G.P.$