We consider the sequence $(u_n)$ defined by $u_0 = 5$ and, for all natural integers $n$:
$$u_{n+1} = 2 + \ln\left(u_n^2 - 3\right)$$
We admit that this sequence is well defined.

\textbf{Part A: Exploitation of Python programs}
\begin{enumerate}
  \item Copy and complete the Python script below so that \texttt{suite(k)} which takes a natural integer $k$ as parameter returns the list of the first $k$ values of the sequence $(u_n)$.

  Remark: We specify that, for any strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$.

\begin{verbatim}
def suite(k):
    L = []
    u = 5
    for i in range(......):
        L.append(u)
        u=............
    return(......)
\end{verbatim}

  \item We executed \texttt{suite(9)} below. Make two conjectures: one on the direction of variation of the sequence $(u_n)$ and another on its possible convergence.

\begin{verbatim}
>>> suite(9)
[ 5, 5.091042453358316, 5.131953749864703,
5.150037910978289, 5.157974010229213, 5.1614456706362954,
5.162962248594583, 5.163624356938671, 5.163913344065642]
\end{verbatim}

  \item We then created the function \texttt{mystere(n)} given below and executed \texttt{mystere(10000)}, which returned 1.\\
  Does this output contradict the conjecture made about the direction of variation of the sequence $(u_n)$? Justify.

\begin{verbatim}
def mystere(n):
    L = suite(n)
    c = 1
    for i in range(n - 1):
        if L[i] > L[i + 1]:
            c = 0
    return c

>>> mystere(10000)
1
\end{verbatim}
\end{enumerate}

\textbf{Part B: Study of the convergence of the sequence $(u_n)$}

We consider the function $g$ defined on $[2; +\infty[$ by:
$$g(x) = 2 + \ln\left(x^2 - 3\right)$$
We admit that $g$ is differentiable on $[2; +\infty[$ and we denote $g'$ its derivative function.

\begin{enumerate}
  \item Prove that the function $g$ is increasing on $[2; +\infty[$.
  \item a. Prove by induction that, for all natural integers $n$:
  $$4 \leqslant u_n \leqslant u_{n+1} \leqslant 6$$
  b. Deduce that the sequence $(u_n)$ converges.
\end{enumerate}

\textbf{Part C: Study of the limit value}

We consider the function $f$ defined on $[2; +\infty[$ by:
$$f(x) = 2 + \ln\left(x^2 - 3\right) - x$$
We admit that $f$ is differentiable on $[2; +\infty[$ and we denote $f'$ its derivative function. We give the following variation table of $f$. No justification is requested.

\begin{center}
\begin{tabular}{|c|ccc|}
\hline
$x$ & 2 & 3 & $+\infty$ \\
\hline
 & & $\ln(6) - 1$ & \\
$f(x)$ & & & \\
 & 0 & & $-\infty$ \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item a. Show that the equation $f(x) = 0$ has exactly two solutions on $[2; +\infty[$ which we will denote $\alpha$ and $\beta$ with $\alpha < \beta$.\\
  b. Give the exact value of $\alpha$ and an approximate value to $10^{-3}$ of $\beta$.
  \item Let $\ell$ be the limit of the sequence $(u_n)$. Justify that $f(\ell) = 0$ and determine $\ell$.
\end{enumerate}