Q. 113.5( 136 Votes )

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

Answer :

Given: Sp = a, Sq = b, and Sr = c

To Prove:

Proof:

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

We know that sum of n terms of an A.P is given by:


where, a = first term
n = nth terms 
d = common difference

Therefore, we have,

Sum of first p terms =


......(1)


Sum of first q terms =


 ......(2)


Sum of first p terms =


.....(3)


Subtracting (2) from (1)





Subtracting (3) from (2)





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.

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