\textbf{(For candidates who have not followed the specialization course)}\\
Let the numerical sequence $(u _ { n })$ defined on $\mathbf{N}$ by:
$$u _ { 0 } = 2 \quad \text { and for every natural number } n , u _ { n + 1 } = \frac { 2 } { 3 } u _ { n } + \frac { 1 } { 3 } n + 1$$

\begin{enumerate}
  \item \begin{enumerate}
    \item[a.] Calculate $u _ { 1 } , u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Approximate values to $10 ^ { - 2 }$ may be given.
    \item[b.] Form a conjecture about the monotonicity of this sequence.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Prove that for every natural number $n$,
$$u _ { n } \leqslant n + 3$$
    \item[b.] Prove that for every natural number $n$,
$$u _ { n + 1 } - u _ { n } = \frac { 1 } { 3 } \left( n + 3 - u _ { n } \right)$$
    \item[c.] Deduce a validation of the previous conjecture.
  \end{enumerate}
  \item We denote by $\left( v _ { n } \right)$ the sequence defined on $\mathbf { N }$ by $v _ { n } = u _ { n } - n$.
  \begin{enumerate}
    \item[a.] Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$.
    \item[b.] Deduce that for every natural number $n$,
$$u _ { n } = 2 \left( \frac { 2 } { 3 } \right) ^ { n } + n$$
    \item[c.] Determine the limit of the sequence $(u _ { n })$.
  \end{enumerate}
  \item For every non-zero natural number $n$, we set:
$$S _ { n } = \sum _ { k = 0 } ^ { n } u _ { k } = u _ { 0 } + u _ { 1 } + \ldots + u _ { n } \quad \text { and } \quad T _ { n } = \frac { S _ { n } } { n ^ { 2 } } .$$
  \begin{enumerate}
    \item[a.] Express $S _ { n }$ as a function of $n$.
    \item[b.] Determine the limit of the sequence $(T _ { n })$.
  \end{enumerate}
\end{enumerate}