Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that

Let a₁ and d be the common difference of the A.P.

Given,

Sum of first p terms =

⇒

Sum of first q terms =

⇒

Sum of first p terms =

⇒

Subtracting (II) from (I)

⇒

⇒

Subtracting (III) from (II)

⇒

⇒

From (IV) and (V)

⇒ pq (p – q) (2br – 2cq) = qr (q – r) (2aq – 2bp)

⇒ p (p – q) (2br – 2cq) = r (q – r) (2aq – 2bp)

⇒ (aqr – bpr) (q – r) = (bpr – cpq) (p – q)

Dividing both sides by pqr

⇒

⇒

⇒

Hence, proved.