Question

If the p^th, q^th and r^th terms of an A.P. are a, b, c respectively, prove that a(q – r) + b (r – p) + c(p – q) = 0.

Answer 1

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

a_p = a, ∴ A + (p – 1)D = a ….. (1) 

a_q = b, ∴ A + (q – 1)D = b ……. (2) 

a_r = c, ∴ A + (r – 1)D = c …….. (3) 

∴ a (q – r) + b (r – p) + c (p – q) = [A + (p – l) D] (q – r) + [A + (q – 1) D] 

(r – p) + [A + (r – 1) D] (p – q) [Using (1), (2) and (3)] 

= (q – r + r – p + p – q)A + [(p – l)(q – r) + (q – l)(r – p) + (r – l)(p – q)]D

= (0) A + (pq – pr – q + r + qr – pq – r + p + pr – p – qr + q)D 

= (0)A + (0)D = 0.