\textbf{Candidates who have followed the specialization course}

A smoker decides to quit smoking. We choose to use the following model:
\begin{itemize}
  \item if they do not smoke on a given day, they do not smoke the next day with a probability of 0.9;
  \item if they smoke on a given day, they smoke the next day with a probability of 0.6.
\end{itemize}

We call $p_{n}$ the probability of not smoking on the $n$-th day after their decision to quit smoking and $q_{n}$, the probability of smoking on the $n$-th day after their decision to quit smoking. We assume that $p_{0} = 0$ and $q_{0} = 1$.

\begin{enumerate}
  \item Calculate $p_{1}$ and $q_{1}$.
  \item We use a spreadsheet to automate the calculation of successive terms of the sequences $(p_{n})$ and $(q_{n})$. A screenshot of this spreadsheet is provided below:

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline{2-5}
\multicolumn{1}{c|}{} & A & B & C & D \\
\hline
1 & $n$ & $p_{n}$ & $q_{n}$ & \\
\hline
2 & 0 & 0 & 1 & \\
\hline
3 & 1 & & & \\
\hline
4 & 2 & & & \\
\hline
5 & 3 & & & \\
\hline
\end{tabular}
\end{center}

Column A contains the values of the natural integer $n$.\\
What formulas can be written in cells B3 and C3 so that by copying them downward, we obtain respectively in columns B and C the successive terms of the sequences $(p_{n})$ and $(q_{n})$?
  \item We define the matrices $M$ and, for every natural integer $n$, $X_{n}$ by

$$M = \left(\begin{array}{ll} 0,9 & 0,4 \\ 0,1 & 0,6 \end{array}\right) \quad \text{and} \quad X_{n} = \binom{p_{n}}{q_{n}}.$$

We admit that $X_{n+1} = M \times X_{n}$ and that, for every natural integer $n$, $X_n = M^{n} \times X_{0}$.\\
We define the matrices $A$ and $B$ by $A = \left(\begin{array}{ll} 0,8 & 0,8 \\ 0,2 & 0,2 \end{array}\right)$ and $B = \left(\begin{array}{cc} 0,2 & -0,8 \\ -0,2 & 0,8 \end{array}\right)$.\\
a) Prove that $M = A + 0,5 B$.\\
b) Verify that $A^{2} = A$, and that $A \times B = B \times A = \left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)$.

We admit in the following that, for every strictly positive natural integer $n$, $A^{n} = A$ and $B^{n} = B$.\\
c) Prove that, for every natural integer $n$, $M^{n} = A + 0,5^{n} B$.\\
d) Deduce that, for every natural integer $n$, $p_{n} = 0,8 - 0,8 \times 0,5^{n}$.\\
e) In the long term, can we assert with certainty that the smoker will quit smoking?
\end{enumerate}