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Question
If the pth,qth and rth terms of an A. P. are P, Q, R respectively, then P(q−r)+Q(r−p)+R(p−q) equals ______.
If the pth,qth and rth terms of an A. P. are P, Q, R respectively, then P(q−r)+Q(r−p)+R(p−q) equals ______.
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Solution
The correct option is B 0
Let a be the first term of an A.P and the common difference d.Tp=a+(p−1)d=PTq=a+(q−1)d=QTr=a+(r−1)d=RNow,P(q−r)+Q(r−p)+R(p−q)=Pq−Pr+Qr−Qp+Rp−Rq=q(P−R)+r(Q−P)+p(R−Q)=q(pd−d−rd+d)+r(qd−d−pd+d)+p(rd−d−qd+d)=d(qp−qr+qr−rp+pr−pq)=0
Let a be the first term of an A.P and the common difference d.
Tp=a+(p−1)d=PTq=a+(q−1)d=QTr=a+(r−1)d=R
Now,
P(q−r)+Q(r−p)+R(p−q)=Pq−Pr+Qr−Qp+Rp−Rq=q(P−R)+r(Q−P)+p(R−Q)=q(pd−d−rd+d)+r(qd−d−pd+d)+p(rd−d−qd+d)=d(qp−qr+qr−rp+pr−pq)=0
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