Question

If p,q,r terms of an A.P. are x,y,z respectively . prove that
x(q-r)+y(r-p)+z(p-q)=0​

Answer 1

Answer:

x=a+(p-1)d

y=a+(q-1)d

z=a+(r-1)d

y-z=(q-r)d

q-r = (y-z)/d

x(q-r)=x(y-z)/d......(1)

Similarly

y(r-p)=y(z-x)/d.....(2)

z(p-q) = z(x-y)/d.....(3)

Add (1) + (2) +(3) you will get

x(q-r) + y(r-p) +z(p-q)= (xy-xz +yz-yx+zx-zy)/d=0