If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first(p+q) terms.

S_p = S_q

Let the first term = a, common difference = d

Therefore, according to the question =

Required. To find S_{p + q}

Now =

According to question

or p[2a + pd – d] = q[2a + qd – d]

or 2ap + p²d – pd = 2aq + q²d – qd

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

or 2a + d(p + q) - d = 0

2a + (p + q – 1)d = 0

Now S_{p + q} =

               = = 0