Question 34
Suppose $x_0,x_1,x_2,\ldots$ is a sequence with $x_0=1000$ and
$$x_n=-\frac{1000}{n}(x_0+x_1+\cdots+x_{n-1})\qquad(n\ge1).$$
Find
$$\frac1{2^2}x_0+\frac12x_1+x_2+2x_3+2^2x_4+\cdots+2^{997}x_{999}+2^{998}x_{1000}.$$
$$x_n=-\frac{1000}{n}(x_0+x_1+\cdots+x_{n-1})\qquad(n\ge1).$$
Find
$$\frac1{2^2}x_0+\frac12x_1+x_2+2x_3+2^2x_4+\cdots+2^{997}x_{999}+2^{998}x_{1000}.$$