In 2020, an influencer on social media has 1000 followers on her profile. The number of followers is modelled as follows: each year, she loses $10\%$ of her followers to which 250 new followers are added.\\
For any natural integer $n$, we denote $u_{n}$ the number of followers on her profile in the year $(2020 + n)$, following this model. Thus $u_{0} = 1000$.

\begin{enumerate}
  \item Calculate $u_{1}$.
  \item Justify that for any natural integer $n$, $u_{n+1} = 0.9 u_{n} + 250$.
  \item The Python function named ``suite'' is defined below. In the context of the exercise, interpret the value returned by suite(10).
\end{enumerate}

\begin{verbatim}
def suite(n) :
    u=1000
    for i in range(n) :
        u=0.9*u+250
    return u
\end{verbatim}

\begin{enumerate}
  \setcounter{enumi}{3}
  \item a. Show, using a proof by induction, that for any natural integer $n$, $u_{n} \leqslant 2500$.\\
b. Prove that the sequence $(u_{n})$ is increasing.\\
c. Deduce from the previous questions that the sequence $(u_{n})$ is convergent.
  \item Let $(v_{n})$ be the sequence defined by $v_{n} = u_{n} - 2500$ for any natural integer $n$.\\
a. Show that the sequence $(v_{n})$ is a geometric sequence with common ratio 0.9 and initial term $v_{0} = -1500$.\\
b. For any natural integer $n$, express $v_{n}$ as a function of $n$ and show that:
$$u_{n} = -1500 \times 0.9^{n} + 2500$$
c. Determine the limit of the sequence $(u_{n})$ and interpret it in the context of the exercise.\\
  \item Write a program that determines in which year the number of followers will exceed 2200.\\
Determine this year.
\end{enumerate}