Question

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

Answer 1

Finding common difference.
Sum of $n$ terms of A.P. is $pn+q{n}^2$ i.e. $S_n=pn+q{n}^2$
For $n=1;S_1=a_1=p+q$
For $n=2;S_2=a_1+a_2=2p+4q$
$a_2=S_2-S_1=2p+4q-(p-q)$
$\Rightarrow a_2=p+3q$

Now, common difference $=a_2-a_1=p+3q-(p+q)=2q$

Final answer: Hence, common difference is $2q$.