\textbf{Exercise 3 — Candidates who have not followed the specialization}

The complex plane is equipped with an orthonormal coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$.\\
For every natural integer $n$, we denote by $A _ { n }$ the point with affix $z _ { n }$ defined by:
$$z _ { 0 } = 1 \quad \text { and } \quad z _ { n + 1 } = \left( \frac { 3 } { 4 } + \frac { \sqrt { 3 } } { 4 } \mathrm { i } \right) z _ { n } .$$
We define the sequence ( $r _ { n }$ ) by $r _ { n } = \left| z _ { n } \right|$ for every natural integer $n$.

\begin{enumerate}
  \item Give the exponential form of the complex number $\frac { 3 } { 4 } + \frac { \sqrt { 3 } } { 4 }$ i.
  \item a. Show that the sequence ( $r _ { n }$ ) is geometric with common ratio $\frac { \sqrt { 3 } } { 2 }$.\\
b. Deduce the expression of $r _ { n }$ as a function of $n$.\\
c. What can be said about the length $\mathrm { O } A _ { n }$ as $n$ tends to $+ \infty$ ?
  \item Consider the following algorithm:
\end{enumerate}

\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables & \begin{tabular}{l}
$n$ natural integer \\
$R$ real $P$ strictly positive real \\
\end{tabular} \\
\hline
Input & Request the value of $P$ \\
\hline
Processing & \begin{tabular}{l}
$R$ takes the value 1 $n$ takes the value 0 \\
While $R > P$ \\
$n$ takes the value $n + 1$ \\
$R$ takes the value $\frac { \sqrt { 3 } } { 2 } R$ \\
End while \\
\end{tabular} \\
\hline
Output & Display $n$ \\
\hline
\end{tabular}
\end{center}

a. What is the value displayed by the algorithm for $P = 0.5$ ?\\
b. For $P = 0.01$ we obtain $n = 33$. What is the role of this algorithm?\\
4. a. Prove that the triangle $\mathrm { O } A _ { n } A _ { n + 1 }$ is right-angled at $A _ { n + 1 }$.\\
b. We admit that $z _ { n } = r _ { n } \mathrm { e } ^ { \frac { i n \pi } { 6 } }$.

Determine the values of $n$ for which $A _ { n }$ is a point on the imaginary axis.\\
c. Complete the figure given in the appendix, to be returned with your work, by representing the points $A _ { 6 } , A _ { 7 } , A _ { 8 }$ and $A _ { 9 }$.\\
Construction lines should be visible.