# 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.

Given. For the A.P.

Sp = Sq

Let the first term = a, common difference = d

Therefore, according to the question = Required. To find Sp + 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 Sp + q = = = 0

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