Consider the numerical sequence $(u_n)$ defined on $\mathbb{N}$ by:

$$u _ { 0 } = 2 \quad \text { and for every natural number } n , \quad u _ { n + 1 } = - \frac { 1 } { 2 } u _ { n } ^ { 2 } + 3 u _ { n } - \frac { 3 } { 2 } .$$

\section*{Part A: Conjecture}
\begin{enumerate}
  \item Calculate the exact values, given as irreducible fractions, of $u _ { 1 }$ and $u _ { 2 }$.
  \item Give an approximate value to $10 ^ { - 5 }$ of the terms $u _ { 3 }$ and $u _ { 4 }$.
  \item Conjecture the direction of variation and the convergence of the sequence $(u_n)$.
\end{enumerate}

\section*{Part B: Validation of conjectures}
Consider the numerical sequence $\left( v _ { n } \right)$ defined for every natural number $n$, by:\\
$v _ { n } = u _ { n } - 3$.

\begin{enumerate}
  \item Show that, for every natural number $n , v _ { n + 1 } = - \frac { 1 } { 2 } v _ { n } ^ { 2 }$.
  \item Prove by induction that, for every natural number $n , - 1 \leqslant v _ { n } \leqslant 0$.
  \item a. Prove that, for every natural number $n , v _ { n + 1 } - v _ { n } = - v _ { n } \left( \frac { 1 } { 2 } v _ { n } + 1 \right)$.\\
b. Deduce the direction of variation of the sequence $\left( v _ { n } \right)$.
  \item Why can we then affirm that the sequence $\left( v _ { n } \right)$ converges?
  \item Let $\ell$ denote the limit of the sequence $(v_n)$. It is admitted that $\ell$ belongs to the interval $[ - 1 ; 0 ]$ and satisfies the equality: $\ell = - \frac { 1 } { 2 } \ell ^ { 2 }$. Determine the value of $\ell$.
  \item Are the conjectures made in Part A validated?
\end{enumerate}