Given: the sum of n terms of an AP is 1/2 (an^2 + bn)

To find: common difference.

Put n = 1 we get
First term = 1/2+ b
Put n = 2 we get
First term + second term = 2 × a + 2 × b
Second term = 3/2(a + b)

Therefore common difference will be
Second term – first term
Common difference = 2a