Question

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

Answer 1

Given,

$S_n=nP+\frac{1}{2}n(n-1)Q$

$S_1=a_1=\left(1\right)P+\frac{1}{2}\left(1\right)(1-1)Q$

$=P$

$S_2=a_1+a_2=\left(2\right)P+\frac{1}{2}\left(2\right)(2-1)Q$

$=2P+Q$

Common difference

$d=a_2-a_1=(2P+Q-P)$

$\therefore d=Q+P.$