A patient must take a dose of 2 mL of a medication every hour.\nWe introduce the sequence $\left( u _ { n } \right)$ such that the term $u _ { n }$ represents the quantity of medication, expressed in mL, present in the body immediately after $n$ doses of medication.\nWe have $u _ { 1 } = 2$ and for every strictly positive natural number $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.

\section*{Part A}
Using this model, a doctor seeks to determine after how many doses of medication the quantity present in the patient's body is strictly greater than 9 mL.

\begin{enumerate}
  \item Calculate the value $u _ { 2 }$.
  \item Show, by induction on $n$, that $u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 }$ for every strictly positive natural number $n$.
  \item Determine $\lim _ { n \rightarrow + \infty } u _ { n }$ and give an interpretation of this result in the context of the exercise.
  \item Let $N$ be a strictly positive natural number, does the inequality $u _ { N } \geq 10$ have solutions? Interpret the result of this question in the context of the exercise.
  \item Determine from how many doses of medication the quantity of medication present in the patient's body is strictly greater than 9 mL. Justify your approach.
\end{enumerate}

\section*{Part B}
Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time.\nFor this purpose, we define the sequence ( $S _ { n }$ ) defined for every strictly positive natural number $n$ by

$$S _ { n } = \frac { u _ { 1 } + u _ { 2 } + \cdots + u _ { n } } { n }$$

We admit that the sequence ( $S _ { n }$ ) is increasing.

\begin{enumerate}
  \item Calculate $S _ { 2 }$.
  \item Show that, for every strictly positive natural number $n$,
\end{enumerate}

$$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n }$$

\begin{enumerate}
  \setcounter{enumi}{2}
  \item Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
  \item The following mystery function is given, written in Python language.
\end{enumerate}

\begin{verbatim}
def mystere(k):
    $\mathrm { n } = 1$
    $\mathrm { s } = 2$
    while $\mathrm { s } < \mathrm { k }$ :
        $\mathrm { n } = \mathrm { n } + 1$
        $\mathrm { s } = 10 - 40 / \mathrm { n } + ( 40 * 0.8 * * \mathrm { n } ) / \mathrm { n }$
    return $n$
\end{verbatim}

In the context of the statement, what does the value returned by the input mystere (9) represent?\nDetermine the value returned by the input mystere (9).

\begin{enumerate}
  \setcounter{enumi}{4}
  \item Justify that this value is strictly greater than 10.
\end{enumerate}