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

In a geometric progression,if the ratio of the sum of the first $5$ terms to the sum of their reciprocals is $49$,and the sum of the first and the third term is $35$,then the first term of this geometric progression is:

In a certain test, there are $n$ questions. In this test, $2^{n-i}$ students gave wrong answers to at least $i$ questions, where $i = 1, 2, \ldots, n$. If the total number of wrong answers given is $2047$, then $n$ is equal to:

The sum of the series $5.05 + 1.212 + 0.29088 + \dots \infty$ is

If $S$ is the sum to infinity of a $G.P.$,whose first term is $a$,then the sum of the first $n$ terms is

If $\frac{a+bx}{a-bx} = \frac{b+cx}{b-cx} = \frac{c+dx}{c-dx}$ and $x \neq 0$,then show that $a, b, c,$ and $d$ are in $G.P.$