5. Using the previous questions, the following result can be established, which is admitted.

For every non-zero natural integer $n$,

$$A ^ { n } = \left( \begin{array} { c c } 
- 2 ^ { n + 1 } + 3 ^ { n + 1 } & 3 \times 2 ^ { n + 1 } - 2 \times 3 ^ { n + 1 } \\
- 2 ^ { n } + 3 ^ { n } & 3 \times 2 ^ { n } - 2 \times 3 ^ { n }
\end{array} \right)$$

Deduce an expression for $u _ { n }$ as a function of $n$.\\
Does the sequence ( $u _ { n }$ ) have a limit?

\section*{APPENDIX for EXERCISE 3, to be returned with the answer sheet}
Graphical representation $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$\\
\includegraphics[max width=\textwidth, alt={}, center]{7ca450dc-ffe7-4eb3-8a4a-b6cc3a7a9c07-7_697_1229_777_449}