Several transmission lines are assembled end to end, and we assume they introduce errors independently of one another. Each line transmits a bit correctly with probability $p$, and incorrectly with probability $1 - p$. We study the transmission of a single bit, which has value 1 at the beginning of transmission. After passing through $n$ transmission lines, we denote:
\begin{itemize}
  \item $p _ { n }$ the probability that the received bit has value 1;
  \item $q _ { n }$ the probability that the received bit has value 0.
\end{itemize}
We therefore have $p _ { 0 } = 1$ and $q _ { 0 } = 0$.\\
We define the following matrices:

$$A = \left( \begin{array} { c c } 
p & 1 - p \\
1 - p & p
\end{array} \right) \quad X _ { n } = \binom { p _ { n } } { q _ { n } } \quad P = \left( \begin{array} { c c } 
1 & 1 \\
1 & - 1
\end{array} \right) .$$

We admit that, for every integer $n$, we have: $X _ { n + 1 } = A X _ { n }$ and therefore, $X _ { n } = A ^ { n } X _ { 0 }$.

\begin{enumerate}
  \item a. Show that $P$ is invertible and determine $P ^ { - 1 }$.\\
b. We set: $D = \left( \begin{array} { c c } 1 & 0 \\ 0 & 2 p - 1 \end{array} \right)$.\\
Verify that: $A = P D P ^ { - 1 }$.\\
c. Show that, for every integer $n \geqslant 1$,
$$A ^ { n } = P D ^ { n } P ^ { - 1 } .$$
d. Using the screenshot of a computer algebra system given below, determine the expression of $q _ { n }$ as a function of $n$.

\begin{center}
\begin{tabular}{ | l | l l | }
\hline
1 & $X 0 : = [ [ 1 ] , [ 0 ] ]$ &  \\
\hline
 & $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$ & M \\
\hline
2 & $\mathrm { P } : = [ [ 1,1 ] , [ 1 , - 1 ] ]$ &  \\
\hline
 & $\left[ \begin{array} { c c } 1 & 1 \\ 1 & - 1 \end{array} \right]$ & M \\
\hline
3 & $\mathrm { D } : = [ [ 1,0 ] , [ 0,2 * p - 1 ] ]$ &  \\
\hline
 & $\left[ \begin{array} { c c } 1 & 0 \\ 0 & 2 * p - 1 \end{array} \right]$ & M \\
\hline
4 & $P * \left( D ^ { \wedge } n \right) * P ^ { \wedge } ( - 1 ) * X 0$ &  \\
\hline
 & $\left[ \frac { ( 2 * p - 1 ) ^ { n } + 1 } { 2 } \right]$ &  \\
 & $\frac { - ( 2 * p - 1 ) ^ { n } + 1 } { 2 }$ & M \\
\hline
\end{tabular}
\end{center}

  \item We assume in this question that $p$ equals 0.98. We recall that the bit before transmission has value 1. We wish the probability that the received bit has value 0 be less than or equal to 0.25. What is the maximum number of such transmission lines that can be aligned?
\end{enumerate}