Q11 of 176 Page 8

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

Sn = qn2 + pn


Taking n = 1, we get


S1 = q(1)2 + p(1)


⇒ S1 = q + p


⇒ a1 = q + p


Taking n = 2, we get


S2 = q(2)2 + p(2)


⇒ S2 = 4q + 2p


∴ a2 = S2 – S1 = 4q + 2p – q - p = 3q + p


Taking n = 3, we get


S3 = q(3)2 + p(3)


⇒ S3 = 9q + 3p


∴ a3 = S3 – S2 = 9q + 3p – 4q – 2p = 5q + p


So, a = q + p,


d = a2 – a1 = 3q + p – (q + p) = 3q + p – q – p = 2q


Hence, the common difference is 2q.


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