5 points With a concern for environmental preservation, Mr. Durand decides to go to work each morning using his bicycle or public transport. If he chooses to take public transport one morning, he takes public transport again the next day with a probability equal to 0.8. If he uses his bicycle one morning, he uses his bicycle again the next day with a probability equal to 0.4. For every non-zero natural number $n$, we denote:
$T _ { n }$ the event ``Mr. Durand uses public transport on the $n$-th day''
$V _ { n }$ the event ``Mr. Durand uses his bicycle on the $n$-th day''
We denote $p _ { n }$ the probability of the event $T _ { n }$,
On the first morning, he decides to use public transport. Thus, the probability of the event $T _ { 1 }$ is $p _ { 1 } = 1$.
Copy and complete the probability tree below representing the situation for the $2 ^ { \mathrm { nd } }$ and $3 ^ { \mathrm { rd } }$ days.
Calculate $p _ { 3 }$
On the $3 ^ { \mathrm { rd } }$ day, Mr. Durand uses his bicycle. Calculate the probability that he took public transport the day before.
Copy and complete the probability tree below representing the situation for the $n$-th and ( $n + 1$ )-th days.
Show that, for every non-zero natural number $n$, $p _ { n + 1 } = 0,2 p _ { n } + 0,6$.
Show by induction that, for every non-zero natural number $n$, we have $$p _ { n } = 0,75 + 0,25 \times 0,2 ^ { n - 1 } .$$
Determine the limit of the sequence ( $p _ { n }$ ) and interpret the result in the context of the exercise.
\section*{Exercise 2}
5 points\\
With a concern for environmental preservation, Mr. Durand decides to go to work each morning using his bicycle or public transport.\\
If he chooses to take public transport one morning, he takes public transport again the next day with a probability equal to 0.8.\\
If he uses his bicycle one morning, he uses his bicycle again the next day with a probability equal to 0.4.\\
For every non-zero natural number $n$, we denote:
\begin{itemize}
\item $T _ { n }$ the event ``Mr. Durand uses public transport on the $n$-th day''
\item $V _ { n }$ the event ``Mr. Durand uses his bicycle on the $n$-th day''
\item We denote $p _ { n }$ the probability of the event $T _ { n }$,
\end{itemize}
On the first morning, he decides to use public transport. Thus, the probability of the event $T _ { 1 }$ is $p _ { 1 } = 1$.
\begin{enumerate}
\item Copy and complete the probability tree below representing the situation for the $2 ^ { \mathrm { nd } }$ and $3 ^ { \mathrm { rd } }$ days.
\item Calculate $p _ { 3 }$
\item On the $3 ^ { \mathrm { rd } }$ day, Mr. Durand uses his bicycle. Calculate the probability that he took public transport the day before.
\item Copy and complete the probability tree below representing the situation for the $n$-th and ( $n + 1$ )-th days.
\item Show that, for every non-zero natural number $n$, $p _ { n + 1 } = 0,2 p _ { n } + 0,6$.
\item Show by induction that, for every non-zero natural number $n$, we have
$$p _ { n } = 0,75 + 0,25 \times 0,2 ^ { n - 1 } .$$
\item Determine the limit of the sequence ( $p _ { n }$ ) and interpret the result in the context of the exercise.
\end{enumerate}