Question

If the sum of n terms of an A.P. is $(pn+q{n}^2)$, where p and q are constants, find the common difference.

Answer 1

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]=pn+q{n}^2$

$S_n=an+n^{2}d-nd=pn+q{n}^2$

$S_n(a-d)n+n^{2}d-nd=pn+q{n}^2$

$S_n=(a-d)n+n^{2}d=pn+q{n}^2$

$\Rightarrow (a-d)=p$ & d=q

$\Rightarrow$ common difference = q