Question

if the sum of first p terms of an A.P. is equal to the sum of first q terms, then show that the sum of its first (p+q) terms is zero where p =/ q

Answer 1

Let us consider an AP whose first the term is a and common difference is d.

Now, acc. to the question
p/2 (2a+(p-1)d) = q/2 (2a+ (q-1)d)

on solving we get that,
2ap + p2d - pd = 2aq + q2d - qd

2ap - 2aq + p2d - q2d -pd + qd = 0

taking common,

2p - 2q/2(a) + (p2 - q2 -p +q)d = 0

2p - 2q/2(a + p-q(p+q-1)d) = 0

2p - 2q/2(a+ (p+q-1)d) = 0

Hence proved.