The sum of n arithmetic means between a and b

Let a and b be two given numbers

Let A1, A2, ______ An be arithmetic means between a and b.

∴ a, A1, A2, _____ An , b are in A.P

Here total number of terms is n + 2 first term is a last term is b.

$\therefore$ Sum of first $n+2$ terms $=\frac{(n+2)}{2}\{a+b\}$

i. e $a+A_{1}+A_{2}+\ldots \ldots \ldots+A_{n}+b=\left(\frac{n+2}{2}\right)(a+b)$

i. e $A_{1}+A_{2}+\ldots \ldots+A_{n}=\frac{(n+2)(a+b)}{2}-a-b$

$=\frac{(n+2)(a+b)}{2}-(a+b)$

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

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

$=(a+b) \frac{n}{2}$

i. e $A_{1}+A_{2}+\ldots \ldots+A_{n}=n\left(\frac{a+b}{2}\right)$

i.e