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.
Let a and d be the first term and the common difference of the A.P. respectively.
Given,
Sum of first p terms = 
Sum of first q terms = 
Sp = Sq
⇒ p [2a + pd – d] = q [2a + qd – d]
⇒ 2ap + p (p – 1)d = 2aq + q (q – 1)d
⇒ 2a (p – q) + d [p(p – 1) – q(q – 1)] = 0
⇒ 2a (p – q) + d [p2 – q2 – (p – q)] = 0
⇒ 2a (p – q) + d [(p + q)(p – q) – (p – q)] = 0
⇒ (p – q) [2a + d (p + q – 1)] = 0
⇒ [ 2a + d (p + q – 1)] = 0
⇒ 
∴ 
⇒ 
⇒ Sp+q = 0
10