\textbf{Exercise 4 — For candidates who have NOT followed the speciality course}\\

We consider the function $f$ defined on $\mathbb{R}$ by:
$$f(x) = \frac{1}{2}x^{2} - x + \frac{3}{2}$$
Let $a$ be a positive real number. We define the sequence $(u_{n})$ by $u_{0} = a$ and, for every natural number $n$: $u_{n+1} = f(u_{n})$. The purpose of this exercise is to study the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$, depending on different values of its first term $u_{0} = a$.

\begin{enumerate}
  \item Using a calculator, conjecture the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$, for $a = 2.9$ then for $a = 3.1$.
  \item In this question, we assume that the sequence $(u_{n})$ converges to a real number $\ell$.\\
a. By noting that $u_{n+1} = \frac{1}{2}u_{n}^{2} - u_{n} + \frac{3}{2}$, show that $\ell = \frac{1}{2}\ell^{2} - \ell + \frac{3}{2}$.\\
b. Show that the possible values of $\ell$ are 1 and 3.
  \item In this question, we take $a = 2.9$.\\
a. Show that $f$ is increasing on the interval $[1; +\infty[$.\\
b. Show by induction that, for every natural number $n$, we have: $1 \leqslant u_{n+1} \leqslant u_{n}$.\\
c. Show that $(u_{n})$ converges and determine its limit.
  \item In this question, we take $a = 3.1$ and we admit that the sequence $(u_{n})$ is increasing.\\
a. Using the previous questions show that the sequence $(u_{n})$ is not bounded above.\\
b. Deduce the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$.\\
c. The following algorithm calculates the smallest rank $p$ for which $u_{p} > 10^{6}$. Copy and complete this algorithm. $P$ is a natural number and $U$ is a real number.
\begin{verbatim}
P <- 0
U.....
Tant que...
    P ......
    U ......
Fin Tant que
\end{verbatim}
\end{enumerate}