If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).

The sum of n terms of an AP is given by

Where a is the first term and d is the common difference

Given that S_p = q and S_q = p

Subtract (i) from (ii) that is (ii) – (i)

Using a² – b² = (a + b)(a – b)

We have to show that S_p+q = -(p + q)

Using (m) and (n)

⇒ S_p+q = S_p + S_q + pqd

= q + p + pqd

Substitute d from (iii)

= (p + q) – 2(p + q)

= -(p + q)

Now we have to find sum of p – q terms that is S_p-q

Using (m) and (n)

Substitute d from (iii)

Hence sum of p – q terms is