A city has a bike-sharing network where two stations A and B are located at the top of a hill. It is admitted that no bikes from other stations arrive at stations A and B. It is observed that for each hour $n$ on average:
$20 \%$ of the bikes present at hour $n - 1$ at station A are still at this station. $60 \%$ of the bikes present at hour $n - 1$ at station A are at station B and the others are in other stations of the network or in circulation.
$10 \%$ of the bikes present at hour $n - 1$ at station B are at station $\mathrm { A } , 30 \%$ are still at station B and the others are in other stations of the network or in circulation.
At the beginning of the day, station A has 50 bikes, station B has 60 bikes.
Part A
After $n$ hours, let $a _ { n }$ denote the average number of bikes present at station A and $b _ { n }$ the average number of bikes present at station B. Let $U _ { n }$ denote the column matrix $\binom { a _ { n } } { b _ { n } }$ and thus $U _ { 0 } = \binom { 50 } { 60 }$.
Determine the matrix $M$ such that $U _ { n + 1 } = M \times U _ { n }$.
Determine $U _ { 1 }$ and $U _ { 2 }$.
After how many hours is there only one bike left in station A?
Part B
The service decides to study the effects of a supply of stations A and B consisting of bringing 30 bikes to station A and 10 bikes to station B after each hour of operation. After $n$ hours, let $\alpha _ { n }$ denote the average number of bikes present at station A and $\beta _ { n }$ the average number of bikes present at station B. Let $V _ { n }$ denote the column matrix $\binom { \alpha _ { n } } { \beta _ { n } }$ and $V _ { 0 } = \binom { 50 } { 60 }$. Under these conditions $V _ { n + 1 } = M \times V _ { n } + R$ with $R = \binom { 30 } { 10 }$.
Let $I$ denote the matrix $\left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $N$ the matrix $I - M$. a. Let $V$ denote a column matrix with two rows. Show that $V = M \times V + R$ is equivalent to $N \times V = R$. b. It is admitted that $N$ is an invertible matrix and that $N ^ { - 1 } = \left( \begin{array} { l l } 1,4 & 0,2 \\ 1,2 & 1,6 \end{array} \right)$. Deduce that $V = \binom { 44 } { 52 }$.
For every natural number $n$, let $W _ { n } = V _ { n } - V$. a. Show that $W _ { n + 1 } = M \times W _ { n }$. b. It is admitted that: for every natural number $n , W _ { n } = M ^ { n } \times W _ { 0 }$, and $$\text{for every natural number } n \geqslant 1 , M ^ { n } = \frac { 1 } { 2 ^ { n - 1 } } \left( \begin{array} { l l } 0,2 & 0,1 \\ 0,6 & 0,3 \end{array} \right) .$$ Calculate, for every natural number $n \geqslant 1 , V _ { n }$ as a function of $n$. c. Does the average number of bikes present in stations A and B tend to stabilize?
A city has a bike-sharing network where two stations A and B are located at the top of a hill. It is admitted that no bikes from other stations arrive at stations A and B.
It is observed that for each hour $n$ on average:
\begin{itemize}
\item $20 \%$ of the bikes present at hour $n - 1$ at station A are still at this station. $60 \%$ of the bikes present at hour $n - 1$ at station A are at station B and the others are in other stations of the network or in circulation.
\item $10 \%$ of the bikes present at hour $n - 1$ at station B are at station $\mathrm { A } , 30 \%$ are still at station B and the others are in other stations of the network or in circulation.
\item At the beginning of the day, station A has 50 bikes, station B has 60 bikes.
\end{itemize}
\section*{Part A}
After $n$ hours, let $a _ { n }$ denote the average number of bikes present at station A and $b _ { n }$ the average number of bikes present at station B. Let $U _ { n }$ denote the column matrix $\binom { a _ { n } } { b _ { n } }$ and thus $U _ { 0 } = \binom { 50 } { 60 }$.
\begin{enumerate}
\item Determine the matrix $M$ such that $U _ { n + 1 } = M \times U _ { n }$.
\item Determine $U _ { 1 }$ and $U _ { 2 }$.
\item After how many hours is there only one bike left in station A?
\end{enumerate}
\section*{Part B}
The service decides to study the effects of a supply of stations A and B consisting of bringing 30 bikes to station A and 10 bikes to station B after each hour of operation.
After $n$ hours, let $\alpha _ { n }$ denote the average number of bikes present at station A and $\beta _ { n }$ the average number of bikes present at station B. Let $V _ { n }$ denote the column matrix $\binom { \alpha _ { n } } { \beta _ { n } }$ and $V _ { 0 } = \binom { 50 } { 60 }$.\\
Under these conditions $V _ { n + 1 } = M \times V _ { n } + R$ with $R = \binom { 30 } { 10 }$.
\begin{enumerate}
\item Let $I$ denote the matrix $\left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $N$ the matrix $I - M$.\\
a. Let $V$ denote a column matrix with two rows. Show that $V = M \times V + R$ is equivalent to $N \times V = R$.\\
b. It is admitted that $N$ is an invertible matrix and that $N ^ { - 1 } = \left( \begin{array} { l l } 1,4 & 0,2 \\ 1,2 & 1,6 \end{array} \right)$. Deduce that $V = \binom { 44 } { 52 }$.
\item For every natural number $n$, let $W _ { n } = V _ { n } - V$.\\
a. Show that $W _ { n + 1 } = M \times W _ { n }$.\\
b. It is admitted that: for every natural number $n , W _ { n } = M ^ { n } \times W _ { 0 }$, and
$$\text{for every natural number } n \geqslant 1 , M ^ { n } = \frac { 1 } { 2 ^ { n - 1 } } \left( \begin{array} { l l } 0,2 & 0,1 \\ 0,6 & 0,3 \end{array} \right) .$$
Calculate, for every natural number $n \geqslant 1 , V _ { n }$ as a function of $n$.\\
c. Does the average number of bikes present in stations A and B tend to stabilize?
\end{enumerate}