If the sum of n terms of an A.P is (pn + qn²), where p and q are constants, find the common difference.

Question

If the sum of n terms of an A.P is (pn + qn2), where p and q are constants, find the common difference.

Philoid · Accepted Answer

For the A.P.

Sn = pn + qn2

Now S1 = p × 1 + q(1)2

S1 = p × q ⇒ T1 = p + q
Also S2 = p × 2 + q(2)2

= 2p + 4q

We have T1 + T2 = 2p + 4q

or T2 = 2p + 4q – T1

or T2 = 2p + 4q – (p + q)
or T2 = p + 3q

Hence common difference = T2 – T1

= p + 3q –(p + q)

= p + 3q – p – q = 2q