\textbf{(Candidates who followed the specialization course)}

Consider the sequence defined by its first term $u _ { 0 } = 3$ and, for every natural number $n$, by

$$u _ { n + 1 } = 2 u _ { n } + 6$$

\begin{enumerate}
  \item Prove that, for every natural number $n$,
$$u _ { n } = 9 \times 2 ^ { n } - 6$$
  \item Prove that, for every integer $n \geqslant 1 , u _ { n }$ is divisible by 6.\\
Define the sequence of integers ( $\nu _ { n }$ ) by, for every natural number $n \geqslant 1 , \nu _ { n } = \frac { u _ { n } } { 6 }$.
  \item Consider the statement: ``for every non-zero natural number $n$, $v _ { n }$ is a prime number''. Indicate whether this statement is true or false by justifying the answer.
  \item a. Prove that, for every integer $n \geqslant 1 , v _ { n + 1 } - 2 v _ { n } = 1$.\\
b. Deduce that, for every integer $n \geqslant 1 , v _ { n }$ and $v _ { n + 1 }$ are coprime.\\
c. Deduce, for every integer $n \geqslant 1$, the GCD of $u _ { n }$ and $u _ { n + 1 }$.
  \item a. Verify that $2 ^ { 4 } \equiv 1 [ 5 ]$.\\
b. Deduce that if $n$ is of the form $4 k + 2$ with $k$ a natural number, then $u _ { n }$ is divisible by 5.\\
c. Is the number $u _ { n }$ divisible by 5 for the other values of the natural number $n$? Justify.
\end{enumerate}