If the sum of n terms of an A.P. is (pn+qn2), where p and q are constants, find the common difference.
It is known that, for an A.P with first term a and common difference d
Sn=n2[2a+(n−1)d]
According to the given condition,
n2[2a+(n−1)d]=pn+qn2⇒n2[2a+nd−d]=pn+qn2⇒na+n2d2−n.d2=pn+qn2
Comparing the coefficient of
n2 on both sides,we obtain
d2=q∴d=2q
Thus, the common difference of the A.P. is 2q.