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.

The sum of n terms of an A.P. is given as:

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

The standard formula for the sum of n terms of an A.P. with first term $a$ and common difference $d$ is:

$S_n = \frac{n}{2}[2a + (n-1)d]$

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Expanding the standard formula:

$S_n = \frac{n}{2} \times 2a + \frac{n}{2} \times (n-1)d$

$S_n = na + \frac{n(n-1)d}{2}$

$S_n = na + \frac{1}{2}n(n-1)d$

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Comparing the given formula with the standard formula:

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

Standard: $S_n = na + \frac{1}{2}n(n-1)d$

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Since both expressions represent $S_n$ for all values of n, the corresponding coefficients must be equal.

The coefficient of $n$ gives: $P = a$

The coefficient of $\frac{1}{2}n(n-1)$ gives: $Q = d$

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Therefore, the common difference $= Q$