\section*{Exercise 1}

We propose to compare the evolution of an animal population in two distinct environments A and B.

On January $1^{\text{st}}$ 2025, 6000 individuals are introduced into each of environments A and B.

\section*{Part A}
In this part, we study the evolution of the population in environment A.\\
We assume that in this environment, the evolution of the population is modelled by a geometric sequence $(u_n)$ with first term $u_0 = 6$ and common ratio 0.93.\\
For every natural number $n$, $u_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.

\begin{enumerate}
  \item Give, according to this model, the population on January $1^{\text{st}}$ 2026.
  \item For every natural number $n$, express $u_n$ as a function of $n$.
  \item Determine the limit of the sequence $(u_n)$.
\end{enumerate}

Interpret this result in the context of the exercise.

\section*{Part B}
In this part, we study the evolution of the population in environment B.\\
We assume that in this environment, the evolution of the population is modelled by the sequence $(v_n)$ defined by

$$v_0 = 6 \text{ and for every natural number } n, v_{n+1} = -0.05 v_n^2 + 1.1 v_n.$$

For every natural number $n$, $v_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.

\begin{enumerate}
  \item Give, according to this model, the population on January $1^{\text{st}}$ 2026.
\end{enumerate}

Let $f$ be the function defined on the interval $[0; +\infty[$ by

$$f(x) = -0.05x^2 + 1.1x$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Prove that the function $f$ is increasing on the interval $[0; 11]$.
  \item Prove by induction that for every natural number $n$, we have
$$2 \leqslant v_{n+1} \leqslant v_n \leqslant 6$$
  \item Deduce that the sequence $(v_n)$ is convergent to a limit $\ell$.
  \item a. Justify that the limit $\ell$ satisfies $f(\ell) = \ell$ then deduce the value of $\ell$.\\
b. Interpret this result in the context of the exercise.
\end{enumerate}

\section*{Part C}
This part aims to compare the evolution of the population in the two environments.

\begin{enumerate}
  \item By solving an inequality, determine the year from which the population of environment A will be strictly less than 3000 individuals.
  \item Using a calculator, determine the year from which the population of environment B will be strictly less than 3000 individuals.
  \item Justify that from a certain year onwards, the population of environment B will exceed the population of environment A.
  \item Consider the Python program opposite.\\
a. Copy and complete this program so that after execution, it displays the year from which the population of environment B is strictly greater than the population of environment A.\\
b. Determine the year displayed after execution of the programme.
\end{enumerate}

\begin{verbatim}
n=0
u=6
v = 6
while...:
    u = ...
    v=...
    n = n+1
print (2025 + n)
\end{verbatim}