Chapter 1.2, Problem 50E

For an arbitrary positive integer $n\ge 3$ , find all solutions $x_1,x_2,x_3,\mathrm{...},x_n$ of the simultaneous equations $x_2=\frac{1}{2}\left(x_1+x_3\right),x_3=\frac{1}{2}\left(x_2+x_4\right),\mathrm{...},x_{n-1}=\frac{1}{2}\left(x_{n-2}+x_n\right)$ .

Note that we are asked to solve the simultaneous equations $x_k=\frac{1}{2}\left(x_{k-1}+x_{k+1}\right)$ ,for $k=2,3,\mathrm{...},n-1$ .