Find the sum up to $20$ terms in the geometric progression $0.15, 0.015, 0.0015, \dots$

For a $G.P.$,if $(m+n)^{\text{th}}$ term is $p$ and $(m-n)^{\text{th}}$ term is $q$,then the $m^{\text{th}}$ term is $.........$

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

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 an increasing geometric progression of positive real numbers. If $A_{1} A_{3} A_{5} A_{7} = \frac{1}{1296}$ and $A_{2} + A_{4} = \frac{7}{36}$,then the value of $A_{6} + A_{8} + A_{10}$ is equal to

Find the sum of the first $n$ terms and the sum of the first $5$ terms of the geometric series $1 + \frac{2}{3} + \frac{4}{9} + \dots$