\textbf{Exercise 3: Sequences}

The population of an endangered species is closely monitored in a nature reserve.\\
Climate conditions as well as poaching cause this population to decrease by $10\%$ each year.\\
To compensate for these losses, 100 individuals are reintroduced into the reserve at the end of each year.\\
We wish to study the evolution of the population size of this species over time. For this, we model the population size of the species by the sequence $(u_n)$ where $u_n$ represents the population size at the beginning of the year $2020 + n$.\\
We admit that for all natural integer $n$, $u_n \geqslant 0$.\\
At the beginning of the year 2020, the studied population has 2000 individuals, thus $u_0 = 2000$.
\begin{enumerate}
  \item Justify that the sequence $(u_n)$ satisfies the recurrence relation:
    $$u_{n+1} = 0.9u_n + 100.$$
  \item Calculate $u_1$ then $u_2$.
  \item Prove by induction that for all natural integer $n$: $1000 < u_{n+1} \leqslant u_n$.
  \item Is the sequence $(u_n)$ convergent? Justify your answer.
  \item We consider the sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 1000$.\\
    a. Show that the sequence $(v_n)$ is geometric with common ratio 0.9.\\
    b. Deduce that, for all natural integer $n$, $u_n = 1000\left(1 + 0.9^n\right)$.\\
    c. Determine the limit of the sequence $(u_n)$.\\
    Give an interpretation of this in the context of this exercise.
  \item We wish to determine the number of years necessary for the population size to fall below a certain threshold $S$ (with $S > 1000$).\\
    a. Determine the smallest integer $n$ such that $u_n \leqslant 1020$.\\
    Justify your answer by a calculation.\\
    b. In the Python program opposite, the variable $n$ denotes the number of years elapsed since 2020, the variable $u$ denotes the population size.\\
    Copy and complete this program so that it returns the number of years necessary for the population size to fall below the threshold $S$.
\begin{verbatim}
def population(S) :
    n=0
    u=2000
    while ......:
        u= ...
        n = ...
    return ...
\end{verbatim}
\end{enumerate}