Find the fractional value of $0.125125125 \dots$

In an increasing geometric series,the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25$. Then,the sum of the $4^{\text{th}}$,$6^{\text{th}}$,and $8^{\text{th}}$ terms is equal to

Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is $4$ and the second term is $3/4$,find the values of $a$ and $r$.

The product of three consecutive terms of a geometric progression is $216$ and the sum of the products of these terms taken two at a time is $156$. Find the terms.

The least positive integer $n$ such that $1 - \frac{2}{3} - \frac{2}{3^2} - \dots - \frac{2}{3^{n-1}} < \frac{1}{100}$ is

$A$ $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of the terms occupying odd places,then the common ratio will be equal to