The sum of (m + n)^th and (m − n)^th terms of an A.P. is equal to twice the m^th term.

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

It is known that the k^th term of an A.P. is given by:

a_k = a + (k − 1)d

∴ a_m+n = a + (m + n − 1)d

a_m−n = a + (m − n − 1)d

a_m = a + (m − 1)d

∴ a_m+n + a_m−n = [a + (m + n − 1)d] + [a + (m − n − 1)d]

= 2a + [(m + n − 1) + (m − n − 1)]d

= 2a + (2m − 2)d

= 2[a + (m − 1)d]

= 2a_m

Hence, the sum of (m + n)^th and (m − n)^th terms of an A.P. is equal to twice the m^th term.