SE-SKOLORNAS 1991 Problem 3

The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$ , $x_{k+1} = [(n - \sum_0^k x_i)/2]$ . Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$ .