The correct option is
C 0We know foe any nth term of an A.P, nth term =a+(n−1)d where a= first term and d= common difference.
∴ a1+(p−1)d=1a,
a1+(9−1)d=1h and a1+(r−1)d=1c
∴ 1a=(a1−d)+pd−−−(1), 1b=(a1−d)+qd−−−(2) and 1c=(a1−d)+rd−−−(3)
By (1)−(2),(2)−(3) and (3)−(1)
1a−1b=(p−q)d,1b,1c=(q−r)d and
1c−1a=(r−p)d
∴ b−ad(ab)=p−q,c−bd(bc)=q−r,a−cd(ac)=r−p
or, b−ad=ab(p−q),c−bd=bc(q−r),a−cd=ac(r−p)
Adding all the equation
(b−a)+(c−b)+(a−c)d=ab(p−q)+bc(q−r)+ac(r−p)
or, ab(p−q)+bc(q−r)+ac(r−p)=0------Proved,