A patient is given a medication by intravenous injection. The amount of medication in the blood decreases as a function of time. The purpose of the exercise is to study, for different hypotheses, the evolution of this amount minute by minute.
An injection of 10 mL of medication is performed at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. For all natural integer $n$, we denote by $u_{n}$ the amount of medication, in mL, remaining in the blood after $n$ minutes. Thus $u_{0} = 10$. a. What is the nature of the sequence $\left(u_{n}\right)$? b. For all natural integer $n$, give the expression of $u_{n}$ as a function of $n$. c. After how much time does the amount of medication remaining in the blood become less than $1\%$ of the initial amount? Justify the answer.
A machine performs an injection of 10 mL of medication at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. When the amount of medication falls below 5 mL, the machine reinjects 4 mL of product. After 15 minutes, the machine is stopped. For all natural integer $n$, we denote by $v_{n}$ the amount of medication, in mL, remaining in the blood at minute $n$. The following algorithm gives the amount of medication remaining minute by minute.
\begin{tabular}{l} Variables:
Initialization:
Processing:
&
$n$ is a natural integer.
$v$ is a real number.
Assign to $v$ the value 10.
For $n$ going from 1 to 15
Assign to $v$ the value $0.8 \times v$.
If $v < 5$ then assign to $v$ the value $v + 4$
Display $v$.
End of loop.
\hline \end{tabular} a. Calculate the missing elements of the table below giving, rounded to $10^{-2}$ and for $n$ greater than or equal to 1, the amount of medication remaining minute by minute obtained with the algorithm.
$n$
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
$v_{n}$
10
8
6.4
8.15
6.52
5.21
8.17
6.54
5.23
8.18
6.55
5.24
b. After 15 minutes, what total amount of medication has been injected into the body? c. We wish to program the machine so that it injects 2 mL of product when the amount of medication in the blood is less than or equal to 6 mL and that it stops after 30 minutes. Rewrite the previous algorithm by modifying it so that it displays the amount of medication, in mL, remaining in the blood minute by minute with this new protocol.
We program the machine so that:
at time 0, it injects 10 mL of medication,
every minute, it injects 1 mL of medication.
It is estimated that $20\%$ of the medication present in the blood is eliminated per minute. For all natural integer $n$, we denote by $w_{n}$ the amount of medication, in mL, present in the blood of the patient after $n$ minutes. a. Justify that for all natural integer $n$, $w_{n+1} = 0.8w_{n} + 1$. b. For all natural integer $n$, we set $z_{n} = w_{n} - 5$. Prove that $(z_{n})$ is a geometric sequence whose ratio and first term we will specify. c. Deduce the expression of $w_{n}$ as a function of $n$. d. What is the limit of the sequence $\left(w_{n}\right)$? What interpretation can be given to this?
A patient is given a medication by intravenous injection. The amount of medication in the blood decreases as a function of time. The purpose of the exercise is to study, for different hypotheses, the evolution of this amount minute by minute.
\begin{enumerate}
\item An injection of 10 mL of medication is performed at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. For all natural integer $n$, we denote by $u_{n}$ the amount of medication, in mL, remaining in the blood after $n$ minutes. Thus $u_{0} = 10$.\\
a. What is the nature of the sequence $\left(u_{n}\right)$?\\
b. For all natural integer $n$, give the expression of $u_{n}$ as a function of $n$.\\
c. After how much time does the amount of medication remaining in the blood become less than $1\%$ of the initial amount? Justify the answer.
\item A machine performs an injection of 10 mL of medication at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. When the amount of medication falls below 5 mL, the machine reinjects 4 mL of product.\\
After 15 minutes, the machine is stopped.\\
For all natural integer $n$, we denote by $v_{n}$ the amount of medication, in mL, remaining in the blood at minute $n$.\\
The following algorithm gives the amount of medication remaining minute by minute.
\begin{center}
\begin{tabular}{|l|l|}
\hline
\begin{tabular}{l}
Variables: \\
Initialization: \\
Processing: \\
\end{tabular} & \begin{tabular}{l}
$n$ is a natural integer. \\
$v$ is a real number. \\
Assign to $v$ the value 10. \\
For $n$ going from 1 to 15 \\
Assign to $v$ the value $0.8 \times v$. \\
If $v < 5$ then assign to $v$ the value $v + 4$ \\
Display $v$. \\
End of loop. \\
\end{tabular} \\
\hline
\end{tabular}
\end{center}
a. Calculate the missing elements of the table below giving, rounded to $10^{-2}$ and for $n$ greater than or equal to 1, the amount of medication remaining minute by minute obtained with the algorithm.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$n$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\
\hline
$v_{n}$ & 10 & 8 & 6.4 & & & & & 8.15 & 6.52 & 5.21 & 8.17 & 6.54 & 5.23 & 8.18 & 6.55 & 5.24 \\
\hline
\end{tabular}
\end{center}
b. After 15 minutes, what total amount of medication has been injected into the body?\\
c. We wish to program the machine so that it injects 2 mL of product when the amount of medication in the blood is less than or equal to 6 mL and that it stops after 30 minutes.\\
Rewrite the previous algorithm by modifying it so that it displays the amount of medication, in mL, remaining in the blood minute by minute with this new protocol.
\item We program the machine so that:
\begin{itemize}
\item at time 0, it injects 10 mL of medication,
\item every minute, it injects 1 mL of medication.
\end{itemize}
It is estimated that $20\%$ of the medication present in the blood is eliminated per minute.\\
For all natural integer $n$, we denote by $w_{n}$ the amount of medication, in mL, present in the blood of the patient after $n$ minutes.\\
a. Justify that for all natural integer $n$, $w_{n+1} = 0.8w_{n} + 1$.\\
b. For all natural integer $n$, we set $z_{n} = w_{n} - 5$.\\
Prove that $(z_{n})$ is a geometric sequence whose ratio and first term we will specify.\\
c. Deduce the expression of $w_{n}$ as a function of $n$.\\
d. What is the limit of the sequence $\left(w_{n}\right)$? What interpretation can be given to this?
\end{enumerate}