Question

Prove that the sum of the n arithmetic means inserted between two quantities is n times the single arithmetic mean between them.

Answer 1

a, $x_1,x_2,..x_n$, b be is an A.P. of $n+2$ terms where $x_1,x_2,...x_n$ are n A.M.'s between a and b
Now $x_1+x_2+x_3+...+x_n$.
$=\frac{n}{2}(x_1+x_n)=\frac{n}{2}(T_2+T_{n+1})$
$=\frac{n}{2}(a+d+b-d)=n\left(\frac{a+b}{2}\right)$
$=$n (sngle A.M. of a and b).