We consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 400$ and for every natural integer $n$:

$$u _ { n + 1 } = 0,9 u _ { n } + 60 .$$

\begin{enumerate}
  \item a. Calculate $u _ { 1 }$ and $u _ { 2 }$.\\
b. Conjecture the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$
  \item Show, by induction, that for every natural integer $n$, we have the inequality
$$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 600$$
  \item a. Show that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent.\\
b. Determine the limit of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify.
  \item A function is given written in Python language:
\begin{verbatim}
def mystere(seuil) :
    n=0
    u=400
    while u <= seuil :
        n = n+1
        u=0.9*u+60
    return n
\end{verbatim}
What value do we obtain by typing in the Python console: mystere(500)?
\end{enumerate}

\textbf{Part B:} A fruit grower owns an orchard where he has room to grow a maximum of 500 trees. Each year he sells $10\%$ of the trees in his orchard and then he plants 60 new trees. The orchard has 400 trees in 2023. The fruit grower thinks he will be able to continue selling and planting trees at the same rate in the coming years. Will he face a space problem in his orchard? Explain your answer.