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

Let the terms of an arithmetic progression be $a_1, a_2, a_3, \dots$. If $\frac{a_1 + a_2 + \dots + a_p}{a_1 + a_2 + \dots + a_q} = \frac{p^2}{q^2}$,where $p \neq q$,then $\frac{a_6}{a_{21}} = \dots$

The first term of an $A.P.$ of consecutive integers is ${p^2} + 1$. The sum of $(2p + 1)$ terms of this series can be expressed as:

Let $T_r$ be the $r$-th term of an arithmetic progression for $r = 1, 2, 3, \dots$. If for some positive integers $m$ and $n$,$T_m = \frac{1}{n}$ and $T_n = \frac{1}{m}$,then $T_{mn} = \dots$

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... $minutes$.

If the sum of the first $n$ terms of an arithmetic progression is $cn^2$,what is the sum of the squares of these $n$ terms?