Question

If p,q and r term if an ap are respectively a,b and c then find the value of a(q-r)+b(r-p)+c(p-q)​

Answer 1

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

p =a=A+(p−1)D=(A−D)+pD (1)

q =b=A+(q−1)D=(A−D)+qD ..(2)

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.