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Question

If 1a,1b,1c are the pth,qth,rth terms respectively of an A.P the the value of ab(p−q)+bc(q−r)+ca(r−p) is

A
1
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B
2
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C
0
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D
2
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Solution

The correct option is C 0
We know foe any nth term of an A.P, nth term =a+(n1)d where a= first term and d= common difference.
a1+(p1)d=1a,
a1+(91)d=1h and a1+(r1)d=1c
1a=(a1d)+pd(1), 1b=(a1d)+qd(2) and 1c=(a1d)+rd(3)
By (1)(2),(2)(3) and (3)(1)
1a1b=(pq)d,1b,1c=(qr)d and
1c1a=(rp)d
bad(ab)=pq,cbd(bc)=qr,acd(ac)=rp
or, bad=ab(pq),cbd=bc(qr),acd=ac(rp)
Adding all the equation
(ba)+(cb)+(ac)d=ab(pq)+bc(qr)+ac(rp)
or, ab(pq)+bc(qr)+ac(rp)=0------Proved,

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