A medication is administered to a patient intravenously.

\section*{Part A: discrete model of the medicinal quantity}
After an initial injection of 1 mg of medication, the patient is placed on an infusion. It is estimated that, every 30 minutes, the patient's body eliminates 10\% of the quantity of medication present in the blood and receives an additional dose of 0.25 mg of the medicinal substance.\\
We study the evolution of the quantity of medication in the blood with the following model: for any natural integer $n$, we denote by $u _ { n }$ the quantity, in mg, of medication in the patient's blood after $n$ periods of thirty minutes. We therefore have $u _ { 0 } = 1$.

\begin{enumerate}
  \item Calculate the quantity of medication in the blood after half an hour.
  \item Justify that, for any natural integer $n$, $u _ { n + 1 } = 0.9 u _ { n } + 0.25$.
  \item a. Show by induction on $n$ that, for any natural integer $n$, $u _ { n } \leqslant u _ { n + 1 } < 5$.\\
b. Deduce that the sequence $(u _ { n })$ is convergent.
  \item It is estimated that the medication is truly effective when its quantity in the patient's blood is greater than or equal to 1.8 mg.\\
a. Copy and complete the script written in Python language below so as to determine after how many periods of thirty minutes the medication begins to be truly effective.
\begin{verbatim}
def efficace():
    u=1
    n=0
    while ......:
        u=......
        n = n+1
    return n
\end{verbatim}
b. What is the value returned by this script? Interpret this result in the context of the exercise.
  \item Let $(v _ { n })$ be the sequence defined, for any natural integer $n$, by $v _ { n } = 2.5 - u _ { n }$.\\
a. Show that $(v _ { n })$ is a geometric sequence and specify its common ratio and first term $(v _ { 0 })$.\\
b. Show that, for any natural integer $n$, $u _ { n } = 2.5 - 1.5 \times 0.9 ^ { n }$.\\
c. The medication becomes toxic when its quantity present in the patient's blood exceeds 3 mg.\\
According to the chosen model, does the treatment present a risk for the patient? Justify.
\end{enumerate}

\section*{Part B: continuous model of the medicinal quantity}
After an initial injection of 1 mg of medication, the patient is placed on an infusion. The flow rate of the medicinal substance administered to the patient is 0.5 mg per hour. The quantity of medication in the patient's blood, as a function of time, is modeled by the function $f$, defined on $[ 0 ; + \infty [$, by
$$f ( t ) = 2.5 - 1.5 \mathrm { e } ^ { - 0.2 t }$$
where $t$ denotes the duration of the infusion expressed in hours.\\
We recall that this medication is truly effective when its quantity in the patient's blood is greater than or equal to 1.8 mg.

\begin{enumerate}
  \item Is the medication truly effective after 3 hours 45 minutes?
  \item According to this model, determine after how much time the medication becomes truly effective.
  \item Compare the result obtained with that obtained in question 4. b. of the discrete model in Part A.
\end{enumerate}