If the sum of n terms of an A.P. is $nP +\frac{1}{2} +n(n-1)Q$, where P and Q are constants, find the common difference.

Q

The correct answer is Option (2) → Q

Given (interpreting the standard A.P. form):

$S_n = nP + \frac{n(n-1)}{2}Q$

(where P, Q are constants)

The nth term of an A.P. is:

$a_n = S_n - S_{n-1}$

$a_n = \left[nP + \frac{n(n-1)}{2}Q\right] - \left[(n-1)P + \frac{(n-1)(n-2)}{2}Q\right]$

$a_n = P + (n-1)Q$

So the common difference:

$d = a_n - a_{n-1} = Q$