\textbf{(For candidates who have followed the specialization course)}\\
We study the population of an imaginary region. On January 1, 2013, this region had 250,000 inhabitants, of which $70 \%$ lived in the countryside and $30 \%$ in the city.\\
Examination of statistical data collected over several years leads to the choice of modelling the population evolution for the coming years as follows:
\begin{itemize}
  \item the total population is globally constant,
  \item each year, $5 \%$ of those living in the city decide to move to the countryside and $1 \%$ of those living in the countryside choose to move to the city.
\end{itemize}
For every natural number $n$, we denote $v _ { n }$ the number of inhabitants of this region living in the city on January 1 of the year $(2013 + n)$ and $c _ { n }$ the number of those living in the countryside on the same date.

\begin{enumerate}
  \item For every natural number $n$, express $v _ { n + 1 }$ and $c _ { n + 1 }$ as functions of $v _ { n }$ and $c _ { n }$.
  \item Let the matrix $A = \left( \begin{array} { l l } 0{,}95 & 0{,}01 \\ 0{,}05 & 0{,}99 \end{array} \right)$.
\end{enumerate}