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Question

If pth,qth and rth term of an A.P are a,b,c respectively, then show that a(qr)+b(rp)+c(pq)=0.

Solution
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Let A=first term of the A.P
and
d=common difference of the A.P
Now,
a=A+(p1)d.............(1)
b=A+(q1)d.............(2)
c=A+(r1)d.............(3)

Subtracting (2) from (1),(3) from (2) and (1) from (3) we get
ab=(pq)d.............(4)
bc=(qr)d.............(5)
ca=(rp)d.............(6)

Multiply (4),(5),(6) by c,a,b respectively we have
c(ab)=c(pq)d.........(7)
a(bc)=a(qr)d.........(8)
b(ca)=b(rp)d.........(9)

Now,
a(qr)d+b(rp)d+c(pq)d=[a(qr)+b(rp)+c(pq)]d=0
Now since d is common difference it should be non zero

Hence,
a(qr)+b(rp)+c(pq)=0

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