In a large French city, electric scooters are made available to users. A company, responsible for maintaining the scooter fleet, checks their condition every Monday.

It is estimated that:
\begin{itemize}
  \item when a scooter is in good condition on a Monday, the probability that it is still in good condition the following Monday is 0.9;
  \item when a scooter is in poor condition on a Monday, the probability that it is in good condition the following Monday is 0.4.
\end{itemize}

We are interested in the condition of a scooter during the inspection phases.\\
Let $n$ be a natural integer. We denote $B_n$ the event ``the scooter is in good condition $n$ weeks after its commissioning'' and $p_n$ the probability of $B_n$.\\
When commissioned, the scooter is in good condition. We therefore have $p_0 = 1$.

\begin{enumerate}
  \item Give $p_1$ and show that $p_2 = 0.85$. You may rely on a weighted tree.
  \item Copy and complete the weighted tree.
  \item Deduce that, for all natural integer $n$, $p_{n+1} = 0.5p_n + 0.4$.
  \item a. Prove by induction that for all natural integer $n$, $p_n \geqslant 0.8$.\\
b. Based on this result, what communication can the company consider to highlight the reliability of the fleet?
  \item a. Consider the sequence $(u_n)$ defined for all natural integer $n$ by $u_n = p_n - 0.8$. Show that $(u_n)$ is a geometric sequence and give its first term and common ratio.\\
b. Deduce the expression of $u_n$ then of $p_n$ as a function of $n$.\\
c. Deduce the limit of the sequence $(p_n)$.
\end{enumerate}