Let A, A + D, A + 2D,…….. are in A.P.
Given pth term = A + (p – 1 )D = a ……………. (1)
qth term = A + (q – 1)D = b …………………. (2)
and rth term = A + (r - 1)D = C …………………(3)
X(q - r) ⇒ a(q – r) = (q – r) [A + (p – 1)D]
X(r – p) ⇒ b(r - p) = (r - p)[A + (q – 1)D] x (p - q)
⇒ c(p - q) = (p - q)[A + (r – 1)D]
By adding we get
⇒ a(q - r) + b(r – p) + c(p – q)
= A[(q - r) + (r - p) + (p - q) + D] [(p - 1)(q – r) + (q – 1 )(r – p) + (r - 1)(p – q)]
= A(0) + D(0) = 0