Exercise 4 (5 points) — Candidates who have followed the specialization course

In a public garden, an artist must install an aquatic artwork. This artwork will consist of two basins A and B as well as a filtering reserve R. Initially, the two basins each contain 100 liters of water. A system of pipes allows the following water transfers to be carried out, every hour and in this order:
\begin{itemize}
  \item first, half of basin A empties into reserve R;
  \item then, three quarters of basin B empty into basin A;
  \item finally, 200 liters of water are added to basin A and 300 liters of water are added to basin B.
\end{itemize}
The quantities of water in the two basins A and B are modeled using two sequences $(a_n)$ and $(b_n)$: for any natural number $n$, we denote by $a_n$ and $b_n$ the quantities of water in hundreds of liters that will be respectively contained in basins A and B after $n$ hours. For any natural number $n$, we denote by $U_n$ the column matrix $U_n = \binom{a_n}{b_n}$. Thus $U_0 = \binom{1}{1}$.

1. Justify that, for any natural number $n$, $U_{n+1} = MU_n + C$ where $M = \left(\begin{array}{cc} 0.5 & 0.75 \\ 0 & 0.25 \end{array}\right)$ and $C = \binom{2}{3}$.

2. Consider the matrix $P = \left(\begin{array}{cc} 1 & 3 \\ 0 & -1 \end{array}\right)$.

a. Calculate $P^2$. Deduce that the matrix $P$ is invertible and specify its inverse matrix.

b. Show that $PMP$ is a diagonal matrix $D$ that you will specify.

c. Calculate $PDP$.

d. Prove by induction that, for any natural number $n$, $M^n = PD^nP$.

It is admitted henceforth that for any natural number $n$, $M^n = \left(\begin{array}{cc} 0.5^n & 3 \times 0.5^n - 3 \times 0.25^n \\ 0 & 0.25^n \end{array}\right)$.

3. Show that the matrix $X = \binom{10}{4}$ satisfies $X = MX + C$.

4. For any natural number $n$, we define the matrix $V_n$ by $V_n = U_n - X$.

a. Show that for any natural number $n$, $V_{n+1} = MV_n$.

b. It is admitted that, for any non-zero natural number $n$, $V_n = M^n V_0$. Show that for any non-zero natural number $n$,
$$U_n = \binom{-18 \times 0.5^n + 9 \times 0.25^n + 10}{-3 \times 0.25^n + 4}.$$

5. a. Show that the sequence $(b_n)$ is increasing and bounded above. Determine its limit.

b. Determine the limit of the sequence $(a_n)$.

c. It is admitted that the sequence $(a_n)$ is increasing. Deduce the capacity of the two basins A and B that must be planned for the feasibility of the project, that is, to avoid any overflow.