At the beginning of 2021, a bird colony had 40 individuals. Observation leads to modeling the population evolution by the sequence $(u_n)$ defined for all natural integers $n$ by:
$$\begin{cases} u _ { 0 } & = 40 \\ u _ { n + 1 } & = 0,008 u _ { n } \left( 200 - u _ { n } \right) \end{cases}$$
where $u _ { n }$ denotes the number of individuals at the beginning of the year $( 2021 + n )$.

\begin{enumerate}
  \item Give an estimate, according to this model, of the number of birds in the colony at the beginning of 2022.
\end{enumerate}

Consider the function $f$ defined on the interval $[ 0 ; 100 ]$ by $f ( x ) = 0,008 x ( 200 - x )$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Solve in the interval $[ 0 ; 100 ]$ the equation $f ( x ) = x$.
  \item a. Prove that the function $f$ is increasing on the interval $[ 0 ; 100 ]$ and draw its variation table.\\
b. By noting that, for all natural integers $n , u _ { n + 1 } = f \left( u _ { n } \right)$, prove by induction that, for all natural integers $n$:
$$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 100$$
c. Deduce that the sequence $(u_n)$ is convergent.\\
d. Determine the limit $\ell$ of the sequence $(u_n)$. Interpret the result in the context of the exercise.
  \item Consider the following algorithm:
\begin{verbatim}
def seuil(p) :
    n=0
    u=40
    while u < p:
        n =n+1
        u=0.008*u*(200-u)
    return(n+2021)
\end{verbatim}
The execution of seuil(100) returns no value. Explain why using question 3.
\end{enumerate}