\textbf{For candidates who have not followed the specialization course}

Let $f$ be the function defined on the interval $[ 0 ; 4]$ by
$$f ( x ) = \frac { 2 + 3 x } { 4 + x }$$

\section*{Part A}
We consider the sequence ( $u _ { n }$ ) defined by:
$$u _ { 0 } = 3 \text { and for all natural integer } n , u _ { n + 1 } = f \left( u _ { n } \right) .$$
It is admitted that this sequence is well defined.

\begin{enumerate}
  \item Calculate $u _ { 1 }$.
  \item Show that the function $f$ is increasing on the interval $[ 0 ; 4 ]$.
  \item Show that for all natural integer $n$,
$$1 \leqslant u _ { n + 1 } \leqslant u _ { n } \leqslant 3$$
  \item a. Show that the sequence ( $u _ { n }$ ) is convergent.\\
b. We call $\ell$ the limit of the sequence ( $u _ { n }$ ); show the equality:
$$\ell = \frac { 2 + 3 \ell } { 4 + \ell }$$
c. Determine the value of the limit $\ell$.
\end{enumerate}

\section*{Part B}
We consider the sequence $\left( v _ { n } \right)$ defined by:
$$v _ { 0 } = 0.1 \text { and for all natural integer } n , v _ { n + 1 } = f \left( v _ { n } \right) .$$

\begin{enumerate}
  \item We give in the Annex the representative curve $\mathscr { C } _ { f }$ of the function $f$ and the line $D$ with equation $y = x$.\\
Place on the $x$-axis by geometric construction the terms $v _ { 1 } , v _ { 2 }$ and $v _ { 3 }$ on the annex, to be returned with the copy.\\
What conjecture can be formulated about the direction of variation and the behavior of the sequence ( $v _ { n }$ ) as $n$ tends to infinity?
  \item a. Show that for all natural integer $n$,
$$1 - v _ { n + 1 } = \left( \frac { 2 } { 4 + v _ { n } } \right) \left( 1 - v _ { n } \right)$$
b. Show by induction that for all natural integer $n$,
$$0 \leqslant 1 - v _ { n } \leqslant \left( \frac { 1 } { 2 } \right) ^ { n }$$
  \item Does the sequence $\left( v _ { n } \right)$ converge? If so, specify its limit.
\end{enumerate}