\textbf{Exercise 1 (7 points)}\\
Theme: probabilities, sequences\\
In a tourist region, a company offers a bicycle rental service for the day.\\
The company has two distinct rental points, point A and point B. Bicycles can be borrowed and returned indifferently at either of the two rental points.\\
It is assumed that the total number of bicycles is constant and that every morning, when the service opens, each bicycle is at point A or point B.\\
According to a statistical study:
\begin{itemize}
  \item If a bicycle is at point A one morning, the probability that it is at point A the next morning is equal to 0.84;
  \item If a bicycle is at point B one morning, the probability that it is at point B the next morning is equal to 0.76.
\end{itemize}
When the service opens on the first morning, the company has placed half of its bicycles at point A and the other half at point B.\\
We consider a bicycle from the company chosen at random.\\
For every positive integer $n$, the following events are defined:
\begin{itemize}
  \item $A _ { n }$ : ``the bicycle is at point A on the $n$-th morning''
  \item $B _ { n }$ : ``the bicycle is at point B on the $n$-th morning''.
\end{itemize}
For every positive integer $n$, we denote by $a _ { n }$ the probability of event $A _ { n }$ and by $b _ { n }$ the probability of event $B _ { n }$. Thus $a _ { 1 } = 0.5$ and $b _ { 1 } = 0.5$.

\begin{enumerate}
  \item Copy and complete the weighted tree that models the situation for the first two mornings.
  \item a. Calculate $a _ { 2 }$.\\
b. The bicycle is at point A on the second morning. Calculate the probability that it was at point B on the first morning. The probability will be rounded to the nearest thousandth.
  \item a. Copy and complete the weighted tree that models the situation for the $n$-th and $(n + 1)$-th mornings.\\
b. Justify that for every positive integer $n$, $a _ { n + 1 } = 0.6 a _ { n } + 0.24$.
  \item Show by induction that, for every positive integer $n$, $a _ { n } = 0.6 - 0.1 \times 0.6 ^ { n - 1 }$.
  \item Determine the limit of the sequence $(a _ { n })$ and interpret this limit in the context of the exercise.
  \item Determine the smallest positive integer $n$ such that $a _ { n } \geqslant 0.599$ and interpret the result obtained in the context of the exercise.
\end{enumerate}