Exercise 2

4 points
We consider the complex numbers $z_{n}$ defined for every integer $n \geqslant 0$ by the value of $z_{0}$, where $z_{0}$ is different from 0 and 1, and the recurrence relation:
$$z_{n+1} = 1 - \frac{1}{z_{n}}$$

1. a. In this question, we assume that $z_{0} = 2$. Determine the numbers $z_{1}, z_{2}, z_{3}, z_{4}, z_{5}$ and $z_{6}$ b. In this question, we assume that $z_{0} = \mathrm{i}$. Determine the algebraic form of the complex numbers $z_{1}, z_{2}, z_{3}, z_{4}, z_{5}$ and $z_{6}$. c. In this question we return to the general case where $z_{0}$ is a given complex number. What can we conjecture about the values taken by $z_{3n}$ according to the values of the natural integer $n$? Prove this conjecture.
2. Determine $z_{2016}$ in the case where $z_{0} = 1 + \mathrm{i}$.
3. Are there values of $z_{0}$ such that $z_{0} = z_{1}$? What can we say about the sequence $(z_{n})$ in this case?

Exercise 4 — Candidates who have not followed the specialization course
We define the sequence of complex numbers $( z _ { n } )$ in the following way: $z _ { 0 } = 1$ and, for every natural integer $n$,
$$z _ { n + 1 } = \frac { 1 } { 3 } z _ { n } + \frac { 2 } { 3 } \mathrm { i } .$$
We place ourselves in a plane with an orthonormal direct coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. For every natural integer $n$, we denote $\mathrm { A } _ { n }$ the point in the plane with affix $z _ { n }$. For every natural integer $n$, we set $u _ { n } = z _ { n } - \mathrm { i }$ and we denote $\mathrm { B } _ { n }$ the point with affix $u _ { n }$. We denote C the point with affix i.

1. Express $u _ { n + 1 }$ as a function of $u _ { n }$, for every natural integer $n$.
2. Prove that, for every natural integer $n$,

$$u _ { n } = \left( \frac { 1 } { 3 } \right) ^ { n } ( 1 - \mathrm { i } ) .$$

1. a. For every natural integer $n$, calculate, as a function of $n$, the modulus of $u _ { n }$. b. Prove that

$$\lim _ { n \rightarrow + \infty } \left| z _ { n } - \mathrm { i } \right| = 0$$
c. What geometric interpretation can be given of this result?
4. a. Let $n$ be a natural integer. Determine an argument of $u _ { n }$. b. Prove that, as $n$ ranges over the set of natural integers, the points $\mathrm { B } _ { n }$ are collinear. c. Prove that, for every natural integer $n$, the point $\mathrm { A } _ { n }$ belongs to the line with reduced equation:
$$y = - x + 1 .$$

It is given that the complex number $i - 3$ is a root of the polynomial $3 x ^ { 4 } + 10 x ^ { 3 } + A x ^ { 2 } + B x - 30$, where $A$ and $B$ are unknown real numbers. Find the other roots.

Find all complex solutions to the equation: $$x^{4} + x^{3} + 2x^{2} + x + 1 = 0.$$

(a) Find all complex solutions of $z^6 = z + \bar{z}$.
(b) For an integer $n > 1$, how many complex solutions does $z^n = z + \bar{z}$ have?

1. Given $\frac { ( 1 - j ) ^ { 2 } } { z } = 1 + \mathrm { i }$ (where i is the imaginary unit), then the complex number $\mathrm { z } =$

1. Given $\frac { ( 1 - j ) ^ { 2 } } { z } = 1 + \mathrm { i }$ (where i is the imaginary unit), then the complex number $\mathrm { z } =$
A. $1 + i$
B. $1 - i$
C. $- 1 + i$
D. $- 1 - \mathrm { i }$

If $z ( 1 + \mathrm { i } ) = 2 \mathrm { i }$ , then $z =$
A. $- 1 - \mathrm { i }$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm { i }$
D. $1 + \mathrm { i }$

If $\bar { z } ( 1 + \mathrm { i } ) = 1 - \mathrm { i }$, then $z =$
A. $1 - \mathrm { i }$
B. $1 + \mathrm { i }$
C. $- i$
D. $i$

3. Given $( 1 - i ) ^ { 2 } z = 3 + 2 i$, then $z =$
A. $- 1 - \frac { 3 } { 2 } i$
B. $- 1 + \frac { 3 } { 2 } i$
C. $- \frac { 3 } { 2 } + i$
D. $- \frac { 3 } { 2 } - i$

3. Given $(1 - i)^2 z = 3 + 2i$, then $z =$
A. $-1 - \frac{3}{2}i$
B. $-1 + \frac{3}{2}i$
C. $-\frac{3}{2} + i$
D. $-\frac{3}{2} - i$

2. If $\mathrm { i } ( 1 - z ) = 1$, then $z + \bar { z } =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$

Given $\frac { Z } { \mathrm { i } } = \mathrm { i } - 1$, then $Z =$ \_\_\_\_

If $\frac { z } { z - 1 } = 1 + \mathrm { i }$ , then $z =$
A. $- 1 - \mathrm { i }$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm{i}$
D. $1 + \mathrm { i }$

Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re}z=1$, then it is necessary that
(1) $\beta\in(-1,0)$
(2) $|\beta|=1$
(3) $\beta\in(1,\infty)$
(4) $\beta\in(0,1)$

Let $w ( \operatorname { Im } w \neq 0 )$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w - \bar { w } z = k ( 1 - z )$, for some real number $k$, is
(1) $\{ z : z \neq 1 \}$
(2) $\{ z : | z | = 1 , z \neq 1 \}$
(3) $\{ z : z = \bar { z } \}$
(4) $\{ z : | z | = 1 \}$

Let $n$ denote the number of solutions of the equation $z ^ { 2 } + 3 \bar { z } = 0$, where $z$ is a complex number. Then the value of $\sum _ { k = 0 } ^ { \infty } \frac { 1 } { n ^ { k } }$ is equal to
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 2

Let $z$ and $w$ be two complex numbers such that $w = z \bar { z } - 2 z + 2 , \left| \frac { z + i } { z - 3 i } \right| = 1$ and $\operatorname { Re } ( w )$ has minimum value. Then, the minimum value of $n \in N$ for which $w ^ { n }$ is real, is equal to $\_\_\_\_$.

Let $S = \left\{ z \in \mathbb { C } : z ^ { 2 } + \bar { z } = 0 \right\}$. Then $\sum _ { z \in S } ( \operatorname { Re } ( z ) + \operatorname { Im } ( z ) )$ is equal to $\_\_\_\_$ .

Let $\mathrm { z } = \mathrm { a } + i b , \mathrm { b } \neq 0$ be complex numbers satisfying $\mathrm { z } ^ { 2 } = \overline { \mathrm { z } } \cdot 2 ^ { 1 - | z | }$. Then the least value of $n \in N$, such that $z ^ { n } = ( z + 1 ) ^ { n }$, is equal to $\_\_\_\_$.

The number of complex numbers $z$, satisfying $| z | = 1$ and $\left| \frac { z } { \bar { z } } + \frac { \bar { z } } { z } \right| = 1$, is :
(1) 4
(2) 8
(3) 10
(4) 6

If $\bar { z }$ denotes the conjugate of $z$, what is the non-zero complex number $z$ that satisfies the equation $z ^ { 2 } = \bar { z }$ and whose argument is between $\frac { \pi } { 2 }$ and $\pi$?
A) $\frac { - 1 } { 2 } + ( \sqrt { 3 } ) \mathrm { i }$
B) $\frac { - 1 } { 2 } + \left( \frac { \sqrt { 3 } } { 2 } \right) \mathrm { i }$
C) $\frac { - \sqrt { 2 } } { 2 } + \left( \frac { 1 } { 2 } \right) \mathrm { i }$
D) $\frac { - \sqrt { 2 } } { 2 } + \left( \frac { \sqrt { 2 } } { 2 } \right) i$
E) $\frac { - \sqrt { 3 } } { 2 } + \left( \frac { 1 } { 2 } \right) \mathrm { i }$

Let z be a complex number satisfying the equality
$$i \cdot z + 1 = 2 ( 1 - \bar { z } )$$
What is the real part of the complex number z?
A) $\frac { 1 } { 6 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 5 } { 6 }$

$4 z - 3 \bar { z } = \frac { 1 - 18 i } { 2 - i }$\ Which of the following is the complex number $z$ that satisfies this equality?\ A) $- 2 + i$\ B) $- 3 + i$\ C) $4 + 2 i$\ D) $3 - 2 i$\ E) $4 - i$

Let $\bar{z}$ be the conjugate of the complex number $z$,
$$\frac { 6 + 2 i } { z } = \bar { z } + i$$
the sum of the complex numbers $z$ that satisfy the equality is what?
A) $1 + 3 i$
B) $2 + i$
C) $3 + 2 i$
D) $4 + i$