We study a group of 3000 athletes who practice either athletics in club A or basketball in club B.\\
In 2023, club A has 1700 members and club B has 1300.\\
We decide to model the number of members of club A and club B respectively by two sequences $(a_{n})$ and $(b_{n})$, where $n$ denotes the rank of the year starting from 2023.\\
The year 2023 corresponds to rank 0. We then have $a_{0} = 1700$ and $b_{0} = 1300$.\\
For our study, we make the following assumptions:
\begin{itemize}
  \item during the study, no athlete leaves the group;
  \item each year, 15\% of the athletes in club A leave this club and join club B;
  \item each year, 10\% of the athletes in club B leave this club and join club A.
\end{itemize}

\begin{enumerate}
  \item Calculate the number of members of each club in 2024.
  \item For all natural integer $n$, determine a relation linking $a_{n}$ and $b_{n}$.
  \item Show that the sequence $(a_{n})$ satisfies the following relation for all natural integer $n$:
$$a_{n+1} = 0{,}75\, a_{n} + 300.$$
  \item a. Prove by induction that for all natural integer $n$, we have:
$$1200 \leqslant a_{n+1} \leqslant a_{n} \leqslant 1700.$$
b. Deduce that the sequence $(a_{n})$ converges.
  \item Let $\left(v_{n}\right)$ be the sequence defined for all natural integer $n$ by $v_{n} = a_{n} - 1200$.\\
a. Prove that the sequence $\left(v_{n}\right)$ is geometric.\\
b. Express $v_{n}$ as a function of $n$.\\
c. Deduce that for all natural integer $n$, $a_{n} = 500 \times 0{,}75^{n} + 1200$.
  \item a. Determine the limit of the sequence $(a_{n})$.\\
b. Interpret the result of the previous question in the context of the exercise.
  \item a. Copy and complete the Python program below so that it returns the smallest value of $n$ from which the number of members of club A is strictly less than 1280.
\begin{verbatim}
def seuil() :
    n = 0
    A = 1700
    while... :
        n=n+1
        A = ...
    return...
\end{verbatim}
b. Determine the value returned when the seuil function is called.
\end{enumerate}