If the sum of P terms of AP is q and the sum of q terms is p the the sum of p+q terms will be

let the first term and common difference of AP is a and d....

sum of first q terms =sum of first p terms =k

q/2{2a+(q-1)d} = p/2{2a+(p-1)d} = k 

2a = 2k(q+p-1)/pq   and d=-2k/pq

now sum of first p+q terms =Sp+q= p+q/2 {2a +(p+q-1)d}

                                     =(p+q)/2{2k(p+q-1)/pq  - 2k(p+q-1)/pq }

                                    =0

Regards

Arun