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

$A$ $G.P.$ 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

Let $n (> 1)$ be a positive integer. The largest integer $m$ such that $(n^m + 1)$ divides $(1 + n + n^2 + \dots + n^{127})$ is:

If $64, 27, 36$ are the $P^{\text{th}}$, $Q^{\text{th}}$, and $R^{\text{th}}$ terms of a $GP$, then $P+2Q$ is equal to

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive numbers. Let the sum of its $6^{\text{th}}$ and $8^{\text{th}}$ terms be $2$ and the product of its $3^{\text{rd}}$ and $5^{\text{th}}$ terms be $\frac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to

Three numbers are selected from the set $\{3^1, 3^2, 3^3, \dots, 3^{20}\}$. Find the number of ways the selected numbers can form an increasing $G.P.$