If the $p^{th}$ term of an $A.P.$ is $q$ and the $q^{th}$ term is $p$,then its $r^{th}$ term will be:

The sums of $n$ terms of three $A.P.'s$ whose first term is $1$ and common differences are $1, 2, 3$ are ${S_1}, {S_2}, {S_3}$ respectively. The true relation is

If $1, \log_{y}x, \log_{z}y, -15\log_{x}z$ are in $A.P.$,then

Let $\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \dots, \frac{1}{x_n}$ ($x_i \neq 0$ for $i = 1, 2, \dots, n$) be in $A.P.$ such that $x_1 = 4$ and $x_{21} = 20$. If $n$ is the least positive integer for which $x_n > 50$,then $\sum_{i=1}^n \left( \frac{1}{x_i} \right)$ is equal to:

Find the $25^{th}$ common term of the following $A.P.'s$:
$S_1 = 1, 6, 11, .....$
$S_2 = 3, 7, 11, .....$

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is the square of an integer. The least possible value of the number of digits of $c$ is