Problem 2. For the numerical sequence $x_{0}, x_{1}, \ldots, x_{n}, x_{n+1}, \ldots$ the relations $2 x_{n}=x_{0}+x_{1}+\cdots+x_{n-1}-x_{n}$ hold for all $n=0,1,2, \ldots$ Find each term $x_{n}$ of such a sequence and the values of the sums $S_{n}=$ $x_{0}+x_{1}+\cdots+x_{n}$.
Solution. We have

$$
x_{n}=(1 / 3)\left(x_{0}+x_{1}+\cdots+x_{n-1}\right) \text {. }
$$

Then $x_{0}-$ any, and by induction

$$
\begin{gathered}
x_{n}=x_{0} 4^{n-1} / 3^{n} \text { for } n \geq 1 \\
S_{n}=x_{0}(4 / 3)^{n}
\end{gathered}
$$

Answer: $x_{n}=x_{0} 4^{n-1} / 3^{n}$ for $n \geq 1, \quad S_{n}=x_{0}(4 / 3)^{n}, x_{0}-$ any.