Candidates who have not followed the specialization course A constant volume of $2200 \mathrm{~m}^{3}$ of water is distributed between two basins A and B. Basin A cools a machine. For reasons of thermal balance, a water current is created between the two basins using pumps. The exchanges between the two basins are modeled as follows:
initially, basin A contains $800 \mathrm{~m}^{3}$ of water and basin B contains $1400 \mathrm{~m}^{3}$ of water;
every day, 15\% of the volume of water present in basin B at the beginning of the day is transferred to basin A;
every day, 10\% of the volume of water present in basin A at the beginning of the day is transferred to basin B. For every natural number $n$, we denote:
$a_{n}$ the volume of water, expressed in $\mathrm{m}^{3}$, contained in basin A at the end of the $n$-th day of operation;
$b_{n}$ the volume of water, expressed in $\mathrm{m}^{3}$, contained in basin B at the end of the $n$-th day of operation.
We therefore have $a_{0} = 800$ and $b_{0} = 1400$.
By what relation between $a_{n}$ and $b_{n}$ is the conservation of the total volume of water in the circuit expressed?
Justify that, for every natural number $n, a_{n+1} = \frac{3}{4} a_{n} + 330$.
The algorithm below makes it possible to determine the smallest value of $n$ from which $a_{n}$ is greater than or equal to 1100. Rewrite this algorithm by completing the missing parts.
Variables
: $n$ is a natural number $a$ is a real number
Initialization
: Assign to $n$ the value 0 Assign to $a$ the value 800
Processing
: While $a < 1100$, do: Assign to $a$ the value . . . Assign to $n$ the value . . . End While
Output
: Display $n$
For every natural number $n$, we denote $u_{n} = a_{n} - 1320$. a. Show that the sequence $(u_{n})$ is a geometric sequence and specify its first term and common ratio. b. Express $u_{n}$ as a function of $n$.
Deduce that, for every natural number $n, a_{n} = 1320 - 520 \times \left(\frac{3}{4}\right)^{n}$. 5. We seek to know if, on a given day, the two basins can have, to the nearest cubic meter, the same volume of water. Propose a method to answer this question.
\section*{Exercise 4 (5 points)}
\textit{Candidates who have not followed the specialization course}
A constant volume of $2200 \mathrm{~m}^{3}$ of water is distributed between two basins A and B.\\
Basin A cools a machine. For reasons of thermal balance, a water current is created between the two basins using pumps.\\
The exchanges between the two basins are modeled as follows:
\begin{itemize}
\item initially, basin A contains $800 \mathrm{~m}^{3}$ of water and basin B contains $1400 \mathrm{~m}^{3}$ of water;
\item every day, 15\% of the volume of water present in basin B at the beginning of the day is transferred to basin A;
\item every day, 10\% of the volume of water present in basin A at the beginning of the day is transferred to basin B.\\
For every natural number $n$, we denote:
\item $a_{n}$ the volume of water, expressed in $\mathrm{m}^{3}$, contained in basin A at the end of the $n$-th day of operation;
\item $b_{n}$ the volume of water, expressed in $\mathrm{m}^{3}$, contained in basin B at the end of the $n$-th day of operation.
\end{itemize}
We therefore have $a_{0} = 800$ and $b_{0} = 1400$.
\begin{enumerate}
\item By what relation between $a_{n}$ and $b_{n}$ is the conservation of the total volume of water in the circuit expressed?
\item Justify that, for every natural number $n, a_{n+1} = \frac{3}{4} a_{n} + 330$.
\item The algorithm below makes it possible to determine the smallest value of $n$ from which $a_{n}$ is greater than or equal to 1100.\\
Rewrite this algorithm by completing the missing parts.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables & : $n$ is a natural number $a$ is a real number \\
\hline
Initialization & : Assign to $n$ the value 0 Assign to $a$ the value 800 \\
\hline
Processing & : While $a < 1100$, do: Assign to $a$ the value . . . Assign to $n$ the value . . . End While \\
\hline
Output & : Display $n$ \\
\hline
\end{tabular}
\end{center}
\item For every natural number $n$, we denote $u_{n} = a_{n} - 1320$.\\
a. Show that the sequence $(u_{n})$ is a geometric sequence and specify its first term and common ratio.\\
b. Express $u_{n}$ as a function of $n$.
\end{enumerate}
Deduce that, for every natural number $n, a_{n} = 1320 - 520 \times \left(\frac{3}{4}\right)^{n}$.\\
5. We seek to know if, on a given day, the two basins can have, to the nearest cubic meter, the same volume of water.\\
Propose a method to answer this question.