If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.

Question

Philoid · Accepted Answer

Let a and b be the first and last terms and

The series be a, A₁, A₂, A₃, ........, A_n, b

So We know, Mean

Mean of A₁ and A_n =

A₁ = a + d

A_n = a - d

Therefore, AM

AM between A₂ and A_n-1

Similarly, it is (a + b)/2 for all such numbers, which is constant

Hence, AM = (a + b)/2