Problem source

The sequence of real numbers $u_1$, $u_2$, $u_3$, $\ldots$ is defined by
\begin{equation*}
u_1=2 \,,
\qquad\text{and} \qquad  u_{n+1} = k - \frac{36}{u_n} 
\quad \text{for } n\ge1,
\tag{$*$}
\end{equation*}
where $k$ is a constant.
\begin{questionparts}
\item Determine the values of $k$ for which the sequence $(*)$ is:
\textbf{(a)} constant;
\textbf{(b)} periodic with period 2;
\textbf{(c)} periodic with period 4.
\item
In the case $k=37$, show that $u_n\ge 2$ for all $n$. Given that  in this
case the sequence $(*)$ converges to a limit
$\ell$, find the value of $\ell$.
\end{questionparts}