\textbf{Exercise 5} (5 points) — Candidates who have NOT followed the specialization course

A biologist wishes to study the evolution of the population of an animal species in a reserve. This population is estimated at 12000 individuals in 2016. The constraints of the natural environment mean that the population cannot exceed 60000 individuals.

\section*{Part A: a first model}
In a first approach, the biologist estimates that the population grows by $5\%$ per year. The annual evolution of the population is thus modelled by a sequence $(v_n)$ where $v_n$ represents the number of individuals, expressed in thousands, in $2016 + n$. We thus have $v_0 = 12$.

\begin{enumerate}
  \item Determine the nature of the sequence $(v_n)$ and give the expression of $v_n$ as a function of $n$.
  \item Does this model meet the constraints of the natural environment?
\end{enumerate}

\section*{Part B: a second model}
The biologist then models the annual evolution of the population by a sequence $(u_n)$ defined by $u_0 = 12$ and, for every natural integer $n$,
$$u_{n+1} = -\frac{1.1}{605}u_n^2 + 1.1\, u_n.$$

\begin{enumerate}
  \item We consider the function $g$ defined on $\mathbb{R}$ by
$$g(x) = -\frac{1.1}{605}x^2 + 1.1\, x$$
a. Justify that $g$ is increasing on $[0;60]$.\\
b. Solve in $\mathbb{R}$ the equation $g(x) = x$.

  \item We note that $u_{n+1} = g(u_n)$.\\
a. Calculate the value rounded to $10^{-3}$ of $u_1$. Interpret.\\
b. Prove by induction that, for every natural integer $n$, $0 \leqslant u_n \leqslant 55$.\\
c. Prove that the sequence $(u_n)$ is increasing.\\
d. Deduce the convergence of the sequence $(u_n)$.\\
e. It is admitted that the limit $\ell$ of the sequence $(u_n)$ satisfies $g(\ell) = \ell$. Deduce its value and interpret it in the context of the exercise.

  \item The biologist wishes to determine the number of years after which the population will exceed 50000 individuals with this second model. He uses the following algorithm:

\begin{center}
\begin{tabular}{ | l | l | }
\hline
\multirow{2}{*}{Variables} & $n$ a natural integer \\
\cline{2-2}
 & $u$ a real number \\
\hline
Processing & $n$ takes the value 0 \\
 & $u$ takes the value 12 \\
 & While $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ \\
 & $\quad u$ takes the value $\ldots\ldots\ldots\ldots\ldots$ \\
$n$ takes the value $\ldots\ldots\ldots\ldots\ldots$ & \\
 & End While \\
\hline
Output & Display $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ \\
\hline
\end{tabular}
\end{center}

Copy and complete this algorithm so that it displays as output the smallest integer $r$ such that $u_r \geqslant 50$.
\end{enumerate}