A company has created a Frequently Asked Questions (``FAQ'') on its website. We study the number of questions asked there each month.

\section*{Part A: First modelling}
In this part, we admit that, each month:
\begin{itemize}
  \item $90\%$ of questions already asked the previous month are kept on the FAQ;
  \item 130 new questions are added to the FAQ.
\end{itemize}
During the first month, 300 questions were asked. To estimate the number of questions, in hundreds, present on the FAQ in the $n$-th month, we model the above situation using the sequence $(u_n)$ defined by:
$$u_1 = 3 \text{ and, for all natural integer } n \geqslant 1, u_{n+1} = 0.9u_n + 1.3.$$

\begin{enumerate}
  \item Calculate $u_2$ and $u_3$ and propose an interpretation in the context of the exercise.
  \item Show by induction that for all natural integer $n \geqslant 1$:
$$u_n = 13 - \frac{100}{9} \times 0.9^n.$$
  \item Deduce that the sequence $(u_n)$ is increasing.
  \item We consider the program opposite, written in Python language.

Determine the value returned by entering \texttt{seuil(8.5)} and interpret it in the context of the exercise.
\begin{verbatim}
def seuil(p) :
    n=1
    u=3
    while u<=p :
        n=n+1
        u=0.9*u+1.3
    return n
\end{verbatim}
\end{enumerate}

\section*{Part B: Another modelling}
In this part, we consider a second modelling using a new sequence $(v_n)$ defined for all natural integer $n \geqslant 1$ by:
$$v_n = 9 - 6 \times \mathrm{e}^{-0.19 \times (n-1)}.$$
The term $v_n$ is an estimate of the number of questions, in hundreds, present in the $n$-th month on the FAQ.
\begin{enumerate}
  \item Specify the values rounded to the nearest hundredth of $v_1$ and $v_2$.
  \item Determine, by justifying the answer, the smallest value of $n$ such that $v_n > 8.5$.
\end{enumerate}

\section*{Part C: Comparison of the two models}
\begin{enumerate}
  \item The company considers that it must modify the presentation of its site when more than 850 questions are present on the FAQ. Of these two modellings, which leads to making this modification the soonest? Justify your answer.
  \item By justifying the answer, for which modelling is there the greatest number of questions on the FAQ in the long term?
\end{enumerate}