In May 2020, a company chose to develop telework. It proposed to its 5000 employees in France to choose between telework and working at the company's premises. In May 2020, only 200 of them chose telework. Each month, since the implementation of this measure, $85\%$ of those who had chosen telework the previous month choose to continue, and each month, 450 additional employees choose telework. The number of company employees working from home is modeled by the sequence $(a_n)$. The term $a_n$ designates an estimate of the number of employees working from home in the $n$-th month after May 2020. Thus $a_0 = 200$.

\textbf{Part A:}
\begin{enumerate}
  \item Calculate $a_1$.
  \item Justify that for every natural number $n$, $a_{n+1} = 0.85a_n + 450$.
  \item Consider the sequence $(v_n)$ defined for every natural number $n$ by: $v_n = a_n - 3000$.\\
  a. Prove that the sequence $(v_n)$ is a geometric sequence with common ratio 0.85.\\
  b. Express $v_n$ as a function of $n$ for every natural number $n$.\\
  c. Deduce that, for every natural number $n$, $a_n = -2800 \times 0.85^n + 3000$.
  \item Determine the number of months after which the number of teleworkers will be strictly greater than 2500, after the implementation of this measure in the company.
\end{enumerate}

\textbf{Part B:}\\
The company's managers modeled the number of employees satisfied with this system using the sequence $(u_n)$ defined by $u_0 = 1$ and, for every natural number $n$,
$$u_{n+1} = \frac{5u_n + 4}{u_n + 2}$$
where $u_n$ denotes the number of thousands of employees satisfied with this new measure after $n$ months following May 2020.
\begin{enumerate}
  \item Prove that the function $f$ defined for all $x \in [0;+\infty[$ by $f(x) = \dfrac{5x+4}{x+2}$ is strictly increasing on $[0;+\infty[$.
  \item a. Prove by induction that for every natural number $n$:
$$0 \leqslant u_n \leqslant u_{n+1} \leqslant 4.$$
  b. Justify that the sequence $(u_n)$ is convergent.
  \item We admit that for every natural number $n$,
$$0 \leqslant 4 - u_n \leqslant 3 \times \left(\frac{1}{2}\right)^n.$$
Deduce the limit of the sequence $(u_n)$ and interpret it in the context of the modeling.
\end{enumerate}