a recursion of averages

let us define continued average of n numbers as $Avg_{x_1,x_2.....,x_n}=\dfrac{Avg_{x_1,x_2.....,x_{n-1}}+x_n}{2}$ with $Avg_{x_1}=x_1$

let us define the average as: $A_{x_1,x_2....,x_n}=\dfrac{x_1+x_2+...+x_n}{n}$ find the condition for $A_{x_1,x_2....,x_n}=Avg_{x_1,x_2.....,x_n},n\geq 3$ \begin{array}{10} (i) & 2^{n-1}(x_1+x_2+....+x_n)&=n(x_1+2x_2+...2^{n-1}x_n)\\ (ii)&2^n(x_1+x_2+....+x_n)&=n(2x_1+2x_2+...+2x_n)\\ (iii)& x_1+x_2+....+x_n&=0\\ (iv)& x_1+x_2+...+x_n&=n \end{array}