If $a_m$ denotes the $m^{th}$ term of an $A.P.$,then $a_m$ =

For a series $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms,
Statement $-1$: The sum of the series is always dependent on the value of $n$,i.e.,whether it is even or odd.
Statement $-2$: The sum of the series is $-\frac{n}{2}$ when $n$ is any even integer.

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

$\frac{1^3 + 2^3 + 3^3 + 4^3 + \dots + 12^3}{1^2 + 2^2 + 3^2 + 4^2 + \dots + 12^2} = $

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

Let the sum of the first $n$ terms of a non-constant $A.P., a_1, a_2, a_3, \dots$ be $S_n = 50n + \frac{n(n - 7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.,$ then the ordered pair $(d, a_{50})$ is equal to