We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 3$ and, for every natural number $n$,

$$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + \frac { 1 } { 2 } n + 1 .$$

\section*{Part A}
This part is a multiple choice questionnaire.\\
For each of the following questions, only one of the four proposed answers is correct.\\
To answer, indicate on your paper the question number and the letter of the chosen answer.\\
No justification is required.\\
A wrong answer, no answer, or multiple answers, neither earn nor lose points.

\begin{enumerate}
  \item The value of $u _ { 2 }$ is equal to:\\
a. $\frac { 11 } { 4 }$\\
b. $\frac { 13 } { 2 }$\\
b. 2.7\\
c. 3.5
  \item The sequence $\left( v _ { n } \right)$ defined, for every natural number $n$, by $v _ { n } = u _ { n } - n$ is:\\
a. arithmetic with common difference $\frac { 1 } { 2 }$\\
b. geometric with common ratio $\frac { 1 } { 2 }$\\
c. constant.\\
d. neither arithmetic nor geometric.
  \item We consider the function below, written incompletely in Python language.\\
$n$ denotes a non-zero natural number.\\
We recall that in Python language ``i in range (n)'' means that i varies from 0 to $n - 1$.

\begin{center}
\begin{tabular}{ | l | l | }
\hline
1 & def terme $( \mathrm { n } )$ \\
2 & $\mathrm { U } = 3$ \\
3 & for i in range(n) : \\
4 & $\ldots \ldots \ldots \ldots \ldots \ldots \ldots$ \\
5 & return U \\
\hline
\end{tabular}
\end{center}

For terme(n) to return the value of $u _ { n }$, we can complete line 4 by:\\
a. $\mathrm { U } = \mathrm { U } / 2 + ( \mathrm { i } + 1 ) / 2 + 1$\\
b. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { n } / 2 + 1$\\
c. $U = U / 2 + ( i - 1 ) / 2 + 1$\\
d. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { i } / 2 + 1$
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item Prove by induction that for every natural number $n$:
$$n \leqslant u _ { n } \leqslant n + 3 .$$
  \item Deduce the limit of the sequence $( u _ { n } )$.
  \item Determine the limit of the sequence $\left( \frac { u _ { n } } { n } \right)$.
\end{enumerate}