Suppose the sum of the first $m$ terms of an arithmetic progression is $n$ and the sum of its first $n$ terms is $m$, where $m \neq n$. Then, the sum of the first $(m+n)$ terms of the arithmetic progression is

If the sum of the first $n$ terms of the series $\sqrt 3  + \sqrt {75}  + \sqrt {243}  + \sqrt {507}  + ……$ is $435\sqrt 3 $ , then $n$ equals

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is $170$ . If there are at least $6$ houses in that row and $a$ is the number of the sixth house, then

If ${S_n} = nP + \frac{1}{2}n(n – 1)Q$, where ${S_n}$ denotes the sum of the first $n$ terms of an $A.P.$, then the common difference is

In an $A.P.,$ if $p^{\text {th }}$ term is $\frac{1}{q}$ and $q^{\text {th }}$ term is $\frac{1}{p},$ prove that the sum of first $p q$ terms is $\frac{1}{2}(p q+1),$ where $p \neq q$