Consider the sequence ( $z _ { n }$ ) of complex numbers defined for all natural number $n$ by:

$$\left\{ \begin{array} { l } 
z _ { 0 } = 0 \\
z _ { n + 1 } = \frac { 1 } { 2 } \mathrm { i } \times z _ { n } + 5
\end{array} \right.$$

In the plane with an orthonormal coordinate system, we denote $M _ { n }$ the point with affixe $z _ { n }$. Consider the complex number $z _ { \mathrm { A } } = 4 + 2 \mathrm { i }$ and A the point in the plane with affixe $z _ { \mathrm { A } }$.

\begin{enumerate}
  \item Let ( $u _ { n }$ ) be the sequence defined for all natural number $n$ by $u _ { n } = z _ { n } - z _ { \mathrm { A } }$.\\
a) Show that, for all natural number $n , u _ { n + 1 } = \frac { 1 } { 2 } \mathrm { i } \times u _ { n }$.\\
b) Prove that, for all natural number $n$:

$$u _ { n } = \left( \frac { 1 } { 2 } \mathrm { i } \right) ^ { n } ( - 4 - 2 \mathrm { i } )$$

  \item Prove that, for all natural number $n$, the points $\mathrm { A } , M _ { n }$ and $M _ { n + 4 }$ are collinear.
\end{enumerate}