Question

if the sum of an ap is equal you the sum of its q terms. prove that the sum of p+q terms of it is equal to 0

Answer 1

equate sum of first p and q terms then write d multiplied by its coefficients on one side and a and its coefficients on the other and cancel (p-q) factor, collect all terms one side and you will get expression similar to that of  p+q term's sum which will give you value of sum as 0.

let the first term and common difference of AP is a and d....sum of first q terms =sum of first p terms =kq/2{2a+(q-1)d} = p/2{2a+(p-1)d} = k 2a = 2k(q+p-1)/pq   and d=-2k/pqnow 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

Hope This helps :)