Suppose
$$
u_{n+1}=\frac1{\lambda-u_n}\tag1
$$
after a few iterations, we see that we get something looking like
$$
u_n=\frac{r_n-s_nu}{p_n-q_nu}\tag2
$$
and that given a form as in $(2)$,
$$
\begin{align}
u_{n+1}
&=\frac1{\lambda-\frac{r_n-s_nu}{p_n-q_nu}}\tag{3a}\\
&=\frac{p_n-q_nu}{\lambda(p_n-q_nu)-(r_n-s_nu)}\tag{3b}\\
&=\frac{p_n-q_nu}{(\lambda p_n-r_n)-(\lambda q_n-s_n)u}\tag{3c}
\end{align}
$$
Therefore,
$$
r_{n+1}=p_n\quad\text{and}\quad s_{n+1}=q_n\tag{4a}
$$
and
$$
p_{n+1}=\lambda p_n-r_n\quad\text{and}\quad q_{n+1}=\lambda q_n-s_n\tag{4b}