25.28. Let $x_{1} \stackrel{n \rightarrow \infty}{=}, x_{2}=b$ and $x_{n+1}=\frac{1}{2}\left(x_{n}+x_{n-1}\right)$ for $n \geqslant 1$. Compute $\lim _{n \rightarrow \infty} x_{n}$.
25.28. Answer: $\lim _{n \rightarrow \infty} x_{n}=\frac{a+2 b}{3}$. We will prove that $x_{n}=\frac{a+2 b}{3}+$ $+\left(\frac{-1}{2}\right)^{n-2} \frac{b-a}{3}$. For $n=1$ and 2, the required equality is easily verified. The relation $x_{n+1}=\frac{1}{2}\left(x_{n}+x_{n-1}\right)$ is also easily verified.