$0.14189189189...$ can be expressed as a rational number.

If $a = \sum_{n=0}^\infty x^n$,$b = \sum_{n=0}^\infty y^n$,and $c = \sum_{n=0}^\infty (xy)^n$,where $|x|, |y| < 1$,then which of the following is true?

If in a $G.P.$ of $64$ terms,the sum of all the terms is $7$ times the sum of the odd terms of the $G.P.$,then the common ratio of the $G.P.$ is equal to

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 $A \cdot H = \dots$

The two geometric means between the numbers $1$ and $64$ are

The sum of the $3^{rd}$ and the $4^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000$. If the first term of this $G.P.$ is positive,then its $7^{th}$ term is