Question

If the sum of $n$ terms of an $A.P.$ is $nP+\frac{1}{2}n(n-1)Q$, where $P$ and $Q$ are constants, then find the common difference.

Answer 1

Finding common difference.
Let $a$ be the first term and $d$ be the common difference of given $A.P.$
Given:
Sum of $n$ terms $S_n=nP+\frac{1}{2}\left(n\right)(n-1)Q$
$S_1=a_1=P,S_2=a_1+a_2=2P+Q$
$a_2=S_2-S_1=P+Q$

Now common difference,
$d=a_2-a_1$
$d=P+Q-P=Q$
Hence, the common difference of given $A.P.$ is equal to $Q$.