[tex]x_{n+1}-1=\frac{1}{2-x_{n}}-1=\frac{x_{n}-1}{2-x_{n}}\Rightarrow \frac{1}{x_{n+1}-1}=\frac{2-x_{n}}{x_{n}-1}=\frac{1}{x_{n}-1}-1[/tex]
[tex]v_{n}=\frac{1}{x_{n}-1}\Rightarrow \left\{\begin{matrix} v_{1}=-2 & \\ v_{n+1}=v_{n}-1 & \end{matrix}\right.\Rightarrow v_{n}=-2+(n-1).(-1)=-n-1[/tex]
[tex]\Rightarrow \frac{1}{x_{n}-1}=-n-1\Rightarrow x_{n}-1=-\frac{1}{n+1}\Rightarrow x_{n}=\frac{n}{n+1}[/tex]