Question

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.

Answer 1

Let $A_1,A_2,A_3,\dots \dots A_n$ be the n AMs inserted between two number a and b.

Then,

$a,A_1,A_2,A_3,\dots \dots A_n,b$ are in A.P

So the mean of a and b

$A.M=\frac{a+b}{2}$

The mean of $A_1$ and $A_n$

$A.M=\frac{a+d+b-d}{2}=\frac{a+b}{2}$

Similarly mean of $A_2$ and $A_{n-1}$

$A.M=\frac{a+2d+b-2d}{2}=\frac{a+b}{2}$

Similarly we observe the means is equidistant from beginning and the end is constant $\frac{a+b}{2}$

The A.M is $\frac{a+b}{2}$