Question

The pth, qth and rh terms of an A.P. are a, b and c respectively.
Show that a(q – r) + b(r-p) + c(p - q) = 0​

Answer 1

Let A be the first term and D the common difference of A.P.

T p

​

=a=A+(p−1)D=(A−D)+pD         (1)

T q

​

=b=A+(q−1)D=(A−D)+qD        ..(2)

T r

​

=c=A+(r−1)D=(A−D)+rD       ..(3)

Here we have got two unknowns A and D which are to be eliminated.

We multiply (1),(2) and (3) by q−r,r−p and p−q respectively and add:

a(q−r)+b(r−p)+c(p−q)

=(A−D)[q−r+r−p+p−q]+D[p(q−r)+q(r−p)+r(p−q)]=0.