\textbf{Exercise 2 — Candidates who have not followed the specialization course}

We place ourselves in an orthonormal frame and, for every natural integer $n$, we define the points $\left( A _ { n } \right)$ by their coordinates $\left( x _ { n } ; y _ { n } \right)$ in the following way:

$$\left\{ \begin{array} { l } x _ { 0 } = - 3 \\ y _ { 0 } = 4 \end{array} \text { and for every natural integer } n : \left\{ \begin{array} { l } x _ { n + 1 } = 0,8 x _ { n } - 0,6 y _ { n } \\ y _ { n + 1 } = 0,6 x _ { n } + 0,8 y _ { n } \end{array} \right. \right.$$

\begin{enumerate}
  \item a. Determine the coordinates of the points $A _ { 0 } , A _ { 1 }$ and $A _ { 2 }$.\\
  b. Copy and complete the following algorithm so that it constructs the points $A _ { 0 }$ to $A _ { 20 }$:
\begin{verbatim}
Variables :
i,x,y,t: real numbers
Initialization :
x takes the value -3
y takes the value 4
Processing :
For i ranging from 0 to 20
    Construct the point with coordinates (x;y)
    t takes the value x
    x takes the value ....
    y takes the value ....
End For
\end{verbatim}
  c. Identify the points $A _ { 0 } , A _ { 1 }$ and $A _ { 2 }$ on the point cloud figure. What appears to be the set to which the points $A _ { n }$ belong for every natural integer $n$?
  \item The purpose of this question is to construct geometrically the points $A _ { n }$ for every natural integer $n$. In the complex plane, we denote, for every natural integer $n$, $z _ { n } = x _ { n } + \mathrm { i } y _ { n }$ the affix of the point $A _ { n }$.\\
  a. Let $u _ { n } = \left| z _ { n } \right|$. Show that, for every natural integer $n , u _ { n } = 5$. What geometric interpretation can be made of this result?\\
  b. We admit that there exists a real number $\theta$ such that $\cos ( \theta ) = 0,8$ and $\sin ( \theta ) = 0,6$. Show that, for every natural integer $n , \mathrm { e } ^ { \mathrm { i } \theta } z _ { n } = z _ { n + 1 }$.\\
  c. Prove that, for every natural integer $n , z _ { n } = \mathrm { e } ^ { \mathrm { i } n \theta } z _ { 0 }$.\\
  d. Show that $\theta + \frac { \pi } { 2 }$ is an argument of the complex number $z _ { 0 }$.\\
  e. For every natural integer $n$, determine, as a function of $n$ and $\theta$, an argument of the complex number $z _ { n }$. Explain, for every natural integer $n$, how to construct the point $A _ { n + 1 }$ from the point $A _ { n }$.
\end{enumerate}