If $64, 27, 36$ are the $P^{\text{th}}$,$Q^{\text{th}}$,and $R^{\text{th}}$ terms of a $GP$,then $P+2Q$ is equal to

In a $GP$ series consisting of positive terms,each term is equal to the sum of the next two terms. Then,the common ratio of this $GP$ series is

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1 = \frac{1}{8}$ and $a_2 \neq a_1$ is the arithmetic mean of the next two terms and $S_n = a_1 + a_2 + \ldots + a_n$,then $S_{20} - S_{18}$ is equal to

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

If the sum of the series $1 + \frac{2}{x} + \frac{4}{x^2} + \frac{8}{x^3} + \dots \infty$ is a finite number,then

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive numbers. Let the sum of its $6^{\text{th}}$ and $8^{\text{th}}$ terms be $2$ and the product of its $3^{\text{rd}}$ and $5^{\text{th}}$ terms be $\frac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to