A patient must take a dose of 2 ml of a medication every hour.\\
We 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. We have $u _ { 1 } = 2$ and for every strictly positive natural integer $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.

\textbf{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 that:
$$u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 } \text { for every strictly positive natural integer } n.$$
  \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 integer. Does the inequality $u _ { N } \geqslant 10$ have solutions? Interpret the result of this question in the context of the exercise.
  \item Determine after 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}

\textbf{Part B}\\
Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time.\\
For this purpose, the sequence ( $S _ { n }$ ) is defined for every strictly positive natural integer $n$ 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 integer $n$,
$$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n } .$$
  \item Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
  \item The following mystery function is given, written in Python language:
\begin{verbatim}
def mystere(k) :
    n = 1
    s =2
    while s<k:
        n = n + l
        s = 10 - 40/n + (40*0.8**n)/n
    return n
\end{verbatim}
In the context of the statement, what does the value returned by the call mystere(9) represent?
  \item Justify that this value is strictly greater than 10.
\end{enumerate}