\textbf{(For candidates who have followed the specialization course)}

An organization offers online foreign language learning. Two levels are presented: beginner or advanced. At the beginning of each month, an internet user can register, unregister or change level.\\
At the beginning of month 0, there were 300 internet users at the beginner level and 450 at the advanced level. Each month, half of the beginners move to the advanced level, the other half remain at the beginner level and half of the advanced users who have completed their training unregister from the site. Each month, 100 new internet users register as beginners and 70 as advanced.\\
This situation is modeled by two sequences of real numbers $( d _ { n } )$ and $( a _ { n } )$. For every natural number $n$, $d _ { n }$ and $a _ { n }$ are respectively approximations of the number of beginners and the number of advanced users at the beginning of month $n$.\\
For every natural number $n$, we denote by $U _ { n }$ the column matrix $\binom { d _ { n } } { a _ { n } }$.\\
We set $d _ { 0 } = 300$, $a _ { 0 } = 450$ and, for every integer $n \geqslant 0$

$$\left\{ \begin{aligned}
d _ { n + 1 } & = \frac { 1 } { 2 } d _ { n } + 100 \\
a _ { n + 1 } & = \frac { 1 } { 2 } d _ { n } + \frac { 1 } { 2 } a _ { n } + 70
\end{aligned} \right.$$

\begin{enumerate}
  \item a. Justify the equality $a _ { n + 1 } = \frac { 1 } { 2 } d _ { n } + \frac { 1 } { 2 } a _ { n } + 70$ in the context of the exercise.\\
b. Determine the matrices $A$ and $B$ such that for every natural number $n$,
$$U _ { n + 1 } = A U _ { n } + B$$
  \item Prove by induction that for every natural number $n \geqslant 1$, we have
$$A ^ { n } = \left( \frac { 1 } { 2 } \right) ^ { n } \left( I _ { 2 } + n T \right) \quad \text { where } T = \left( \begin{array} { l l } 
0 & 0 \\
1 & 0
\end{array} \right) \quad \text { and } I _ { 2 } = \left( \begin{array} { l l } 
1 & 0 \\
0 & 1
\end{array} \right) .$$
  \item a. Determine the matrix $C$ that satisfies the equality $C = A C + B$.\\
b. For every integer $n \geqslant 0$, we set $V _ { n } = U _ { n } - \binom { 200 } { 340 }$.
\end{enumerate}