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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
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