Q. 104.0( 134 Votes )

# 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.

Answer :

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 =

S_{p} = S_{q}

⇒ 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 [p^{2} – q^{2} – (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

⇒

∴

⇒

⇒ S_{p+q} = 0

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