We consider the sequence $\left(u_n\right)$ defined for every integer $n \geqslant 0$ by: $\left\{\begin{array}{l}u_{n+1} = 3 - \dfrac{10}{u_n + 4}\\u_0 = 5\end{array}\right.$

\section*{Part A:}
\begin{enumerate}
  \item Determine the exact value of $u_1$ and $u_2$.
  \item Prove by induction that for every natural integer $n$, $u_n \geqslant 1$.
  \item Prove that, for every natural integer $n$, $u_{n+1} - u_n = \dfrac{\left(1 - u_n\right)\left(u_n + 2\right)}{u_n + 4}$.
  \item Deduce the direction of variation of the sequence $(u_n)$.
  \item Justify that the sequence $\left(u_n\right)$ converges.
\end{enumerate}

\section*{Part B:}
We consider the sequence $(v_n)$ defined for every natural integer $n$ by $v_n = \dfrac{u_n - 1}{u_n + 2}$.

\begin{enumerate}
  \item a. Prove that $\left(v_n\right)$ is a geometric sequence whose common ratio and first term $v_0$ we will determine.\\
b. Express $v_n$ as a function of $n$.

Deduce that for every natural integer $n$, $v_n \neq 1$.\\
  \item Prove that for every natural integer $n$, $u_n = \dfrac{2v_n + 1}{1 - v_n}$.\\
  \item Deduce the limit of the sequence $(u_n)$.
\end{enumerate}

\section*{Part C:}
We consider the algorithm below.

\begin{center}
\begin{tabular}{ | l | }
\hline
$u \leftarrow 5$ \\
$n \leftarrow 0$ \\
While $u \geqslant 1.01$ \\
$n \leftarrow n + 1$ \\
$u \leftarrow 3 - \dfrac{10}{u + 4}$ \\
End While \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item After execution of the algorithm, what value is contained in the variable $n$?
  \item Using parts A and B, interpret this value.
\end{enumerate}