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

For $p, q \in R$,consider the real-valued function $f(x) = (x - p)^2 - q$,where $x \in R$ and $q > 0$. Let $a_1, a_2, a_3, a_4$ be in an arithmetic progression with mean $p$ and a positive common difference $d$. If $|f(a_i)| = 500$ for all $i = 1, 2, 3, 4$,then the absolute difference between the roots of $f(x) = 0$ is:

There are $15$ terms in an arithmetic progression. Its first term is $5$ and their sum is $390$. The middle term is

If $\frac{1}{p + q}, \frac{1}{r + p}, \frac{1}{q + r}$ are in $A.P.$,then

Given that $n$ $A$.$M$.'s are inserted between two sets of numbers $a, 2b$ and $2a, b$,where $a, b \in R$. Suppose further that the $m^{th}$ mean between these sets of numbers is the same,then the ratio $a:b$ equals

How many multiples of $5$ are there from $10$ to $95$,including both $10$ and $95$?