The sum of $n$ arithmetic means between $a$ and $b$, is

Let, a and b two given numbers.

Let,

$A_1,A_2,A_3.............A_n$ be Arithmetic means between a and b.

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

$a+A_1+A_2+A_3+......A_n+b=\left(\frac{n+2}{2}\right)\left(a+b\right)$      (first term is =a, last term=b, number of term=n+2)

$A_1,A_2,A_3.............A_n=\frac{\left(n+2\right)\left(a+b\right)}{2}-(a+b)$

 $=\left(a+b\right)\left[\frac{n+2}{2}-1\right]$

 $=\left(a+b\right)\left[\frac{n+2-2}{2}\right]$

 $=n\left(\frac{a+b}{2}\right)$

$=$ n times the single A.M. Between a and b

Hence, this is the answer.