    Question

# If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p+q terms will be

A

0

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B

p - q

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C

p + q

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D

- (p + q)

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Solution

## The correct option is D: −(p+q)We know that the sum of ′n′ terms of an A.P is given by Sn=n2[2a+(n−1)d]Sum of ′p′ terms is given Sp=q ⇒p2[2a+(p−1)d]=q ⇒2ap+(p−1)pd=2q....(i) And, Sum of ′q′ terms is given asSq=p ⇒q2{2a+(q−1)d}=p ⇒2aq+(q−1)qd=2p....(ii) on Subtracting eq.(ii) from eq.(i), we get2ap+(p−1)pd−[2aq+(q−1)qd]=2q−2p⇒2ap+(p−1)pd−2aq−(q−1)qd=2q−2p⇒2a(p−q)+p2d−pd−q2d+qd=−2(p−q)⇒2a(p−q)+p2d−q2d−(p−q)d=−2(p−q)⇒2a(p−q)+(p2−q2)d−(p−q)d=−2(p−q)⇒2a(p−q)+(p−q)(p+q)d−(p−q)d=−2(p−q) [∵(a2−b2)=(a−b)(a+b)]⇒2a+(p+q)d−d=−2 [cancelling (p−q) from both sides]⇒2a+(p+q−1)d=−2……(iii)Now, we have to fing sum of (p+q) termsSp+q=(p+q)2[2a+(p+q−1)d] =(p+q)2[−2] [Using eq.(iii)] =−(p+q)  Suggest Corrections  47      Similar questions  Explore more