Question

The p^th, q^th and r^th terms of an A.P. are a, b, c respectively. Show that

Answer 1

Let t and d be the first term and the common difference of the A.P. respectively.

The n^th term of an A.P. is given by, a_n = t + (n – 1) d

Therefore,

a_p = t + (p – 1) d = a … (1)

a_q = t + (q – 1)d = b … (2)

a_r = t + (r – 1) d = c … (3)

Subtracting equation (2) from (1), we obtain

(p – 1 – q + 1) d = a – b

⇒ (p – q) d = a – b

Subtracting equation (3) from (2), we obtain

(q – 1 – r + 1) d = b – c

⇒ (q – r) d = b – c

Equating both the values of d obtained in (4) and (5), we obtain

Thus, the given result is proved.