If $a = r + r^2 + r^3 + \dots + \infty$,then the value of $r$ is .......

If $x, y, z$ are in geometric progression and $a^x = b^y = c^z$,then . . . . . .

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

What is the sum of the first $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$?

The solution of the equation $1 + a + a^2 + a^3 + \dots + a^x = (1 + a)(1 + a^2)(1 + a^4)$ is given by $x$ is equal to

If $a_1, a_2, a_3, \ldots, a_{10}$ is a geometric progression and $\frac{a_3}{a_1}=25$,then $\frac{a_9}{a_5}$ equals