Biologists are studying the evolution of an insect population in a botanical garden. At the beginning of the study, the population is 100,000 insects. To preserve the balance of the natural environment, the number of insects must not exceed 400,000.

\section*{Part A: Study of a first model in the laboratory}
Observation of the evolution of these insect populations in the laboratory, in the absence of any predator, shows that the number of insects increases by $60\%$ each month. Taking this observation into account, biologists model the evolution of the insect population using a sequence $(u_n)$ where, for every natural number $n$, $u_n$ models the number of insects, expressed in millions, after $n$ months. We therefore have $u_0 = 0.1$.

\begin{enumerate}
  \item Justify that for every natural number $n$: $u_n = 0.1 \times 1.6^n$.
  \item Determine the limit of the sequence $(u_n)$.
  \item By solving an inequality, determine the smallest natural number $n$ from which $u_n > 0.4$.
  \item According to this model, would the balance of the natural environment be preserved? Justify your answer.
\end{enumerate}

\section*{Part B: Study of a second model}
Taking into account the constraints of the natural environment in which the insects evolve, biologists choose a new model. They model the number of insects using the sequence $(v_n)$, defined by:
$$v_0 = 0.1 \text{ and, for every natural number } n, v_{n+1} = 1.6v_n - 1.6v_n^2,$$
where, for every natural number $n$, $v_n$ is the number of insects, expressed in millions, after $n$ months.

\begin{enumerate}
  \item Determine the number of insects after one month.
  \item We consider the function $f$ defined on the interval $\left[0; \frac{1}{2}\right]$ by
$$f(x) = 1.6x - 1.6x^2.$$
a. Solve the equation $f(x) = x$.\\
b. Show that the function $f$ is increasing on the interval $\left[0; \frac{1}{2}\right]$.
  \item a. Show by induction that, for every natural number $n$, $0 \leqslant v_n \leqslant v_{n+1} \leqslant \frac{1}{2}$.\\
b. Show that the sequence $(v_n)$ is convergent.\\
We denote by $\ell$ the value of its limit. We admit that $\ell$ is a solution of the equation $f(x) = x$.\\
c. Determine the value of $\ell$.\\
According to this model, will the balance of the natural environment be preserved? Justify your answer.
  \item The threshold function is given below, written in Python language.\\
a. What do we observe if we enter \texttt{seuil(0.4)}?\\
b. Determine the value returned by entering \texttt{seuil(0.35)}. Interpret this value in the context of the exercise.
\begin{verbatim}
def seuil(a) :
    v=0.1
    n=0
    while v<a :
        v=1.6*v-1.6*v*v
        n=n+1
    return n
\end{verbatim}
\end{enumerate}