If m^th term of an A.P. is 1/n and n^th term is 1/m, then find the sum of its first mn terms.

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

Given: m^th term = a_m = a + (m - 1)d = 1/n ……… (1)

n^th term = a_n = a + (n - 1)d = 1/m ……… (2)

Subtract equation (1) from (2):

a + (m - 1)d - a - (n - 1)d = 1/n - 1/m

⇒ (m - n)d = (m - n)/mn

⇒ d = 1/mn

Put the value of d in equation (1):

a + (m - 1)(1/mn) = 1/n

⇒ a + (1/n) – (1/mn) = 1/n

⇒ a = 1/mn

Sum of first mn terms: