We consider the sequence of complex numbers $(z_n)$ defined by $z_0 = \sqrt{3} - i$ and for every natural number $n$:
$$z_{n+1} = (1 + \mathrm{i})z_n.$$
Parts $A$ and $B$ can be treated independently.

Part A

For every natural number $n$, we set $u_n = |z_n|$.

1. Calculate $u_0$.
2. Prove that $(u_n)$ is a geometric sequence with common ratio $\sqrt{2}$ and first term 2.
3. For every natural number $n$, express $u_n$ as a function of $n$.
4. Determine the limit of the sequence $(u_n)$.
5. Given a positive real number $p$, we wish to determine, using an algorithm, the smallest value of the natural number $n$ such that $u_n > p$. Copy the algorithm below and complete it with the processing and output instructions, so as to display the sought value of the integer $n$. \begin{verbatim} Variables : u is a real number p is a real number n is an integer Initialization : Assign to n the value 0 Assign to u the value 2 Input : Request the value of p Processing : Output : \end{verbatim}

Part B

1. Determine the algebraic form of $z_1$.
2. Determine the exponential form of $z_0$ and of $1 + \mathrm{i}$. Deduce the exponential form of $z_1$.
3. Deduce from the previous questions the exact value of $\cos\left(\frac{\pi}{12}\right)$.

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 $n$.

1. Give the exponential form of the complex number $\frac { 3 } { 4 } + \frac { \sqrt { 3 } } { 4 }$ i.
2. 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 $+ \infty$ ?
3. Consider the following algorithm:

Variables	\begin{tabular}{l} $n$ natural integer
$R$ real $P$ strictly positive real

\hline Input & Request the value of $P$ \hline Processing &

$R$ 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

\hline Output & Display $n$ \hline \end{tabular}
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.

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

1. 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 $n$?
2. 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 $A _ { n }$.

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 $z _ { \mathrm { A } }$.

1. 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 $n$:
   $$u _ { n } = \left( \frac { 1 } { 2 } \mathrm { i } \right) ^ { n } ( - 4 - 2 \mathrm { i } )$$
   
2. Prove that, for all natural number $n$, the points $\mathrm { A } , M _ { n }$ and $M _ { n + 4 }$ are collinear.

Questions 1. and 2. of this exercise may be treated independently. We consider the sequence of complex numbers $( z _ { n } )$ defined for all natural integer $n$ by
$$z _ { n } = \frac { 1 + \mathrm { i } } { ( 1 - \mathrm { i } ) ^ { n } } .$$
We place ourselves in the complex plane with origin O.

1. For all natural integer $n$, we denote $A _ { n }$ the point with affix $z _ { n }$. a. Prove that, for all natural integer $n , \frac { z _ { n + 4 } } { z _ { n } }$ is real. b. Prove then that, for all natural integer $n$, the points O , $A _ { n }$ and $A _ { n + 4 }$ are collinear.
2. For which values of $n$ is the number $z _ { n }$ real?

The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). Consider the sequence of complex numbers ( $z _ { n }$ ) defined by: $$z _ { 0 } = 0 \text { and for every natural number } n , z _ { n + 1 } = ( 1 + \mathrm { i } ) z _ { n } - \mathrm { i }$$ For every natural number $n$, let $A _ { n }$ denote the point with affix $z _ { n }$. Let B denote the point with affix 1.

1. a. Show that $z _ { 1 } = - \mathrm { i }$ and that $z _ { 2 } = 1 - 2 \mathrm { i }$. b. Calculate $z _ { 3 }$. c. On your answer sheet, plot the points $\mathrm { B } , A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ in the direct orthonormal coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. d. Prove that the triangle $\mathrm { B } A _ { 1 } A _ { 2 }$ is isosceles right-angled.
2. For every natural number $n$, set $u _ { n } = \left| z _ { n } - 1 \right|$. a. Prove that for every natural number $n$, we have $u _ { n + 1 } = \sqrt { 2 } u _ { n }$. b. Determine from which natural number $n$ the distance $\mathrm { B } A _ { n }$ is strictly greater than 1000. Detail the approach chosen.
3. a. Determine the exponential form of the complex number $1 + \mathrm { i }$. b. Prove by induction that for every natural number $n$, $z _ { n } = 1 - ( \sqrt { 2 } ) ^ { n } \mathrm { e } ^ { \mathrm { i } \frac { n \pi } { 4 } }$. c. Does the point $A _ { 2020 }$ belong to the x-axis? Justify.