Exercise 4 — Candidates who have not followed the specialization course
For each of the following statements, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. An absence of an answer is not penalized.
Let $\left( u _ { n } \right)$ be the sequence defined by $$u _ { 0 } = 4 \text { and for all natural integer } n , u _ { n + 1 } = - \frac { 2 } { 3 } u _ { n } + 1$$ and let $( \nu _ { n } )$ be the sequence defined by $$\text { for all natural integer } n , v _ { n } = u _ { n } - \frac { 2 } { 3 }$$ Statement 1: The sequence $\left( v _ { n } \right)$ is a geometric sequence.
Let $( w _ { n } )$ be the sequence defined by, for all non-zero natural integer $n$, $$w _ { n } = \frac { 3 + \cos ( n ) } { n ^ { 2 } } .$$ Statement 2: The sequence $\left( w _ { n } \right)$ converges to 0.
Consider the following algorithm: $$\begin{aligned}
& U \leftarrow 5 \\
& N \leftarrow 0
\end{aligned}$$ While $U \leqslant 5000$ $$\begin{aligned}
& U \leftarrow 3 \times U - 8 \\
& N \leftarrow N + 1
\end{aligned}$$ End While Statement 3: At the end of execution, the variable $U$ contains the value 5000.
We denote $\mathbb { C }$ the set of complex numbers. We consider the equation (E) with unknown $z$ in $\mathbb { C }$ $$( z - \mathrm { i } ) \left( z ^ { 2 } + z \sqrt { 3 } + 1 \right) = 0$$ Statement 4: All solutions of equation (E) have modulus 1.
We consider the complex numbers $z _ { n }$ defined by $$z _ { 0 } = 2 \text { and for all natural integer } n , z _ { n + 1 } = 2 \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 2 } } z _ { n } .$$ We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). For all natural integer $n$, we denote $M _ { n }$ the point with affixe $z _ { n }$. Statement 5: For all natural integer $n$, the point O is the midpoint of the segment $\left[ M _ { n } M _ { n + 2 } \right]$.
\section*{Exercise 4 — Candidates who have not followed the specialization course}
For each of the following statements, indicate whether it is true or false, by justifying the answer.\\
One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. An absence of an answer is not penalized.
\begin{enumerate}
\item Let $\left( u _ { n } \right)$ be the sequence defined by
$$u _ { 0 } = 4 \text { and for all natural integer } n , u _ { n + 1 } = - \frac { 2 } { 3 } u _ { n } + 1$$
and let $( \nu _ { n } )$ be the sequence defined by
$$\text { for all natural integer } n , v _ { n } = u _ { n } - \frac { 2 } { 3 }$$
Statement 1: The sequence $\left( v _ { n } \right)$ is a geometric sequence.
\item Let $( w _ { n } )$ be the sequence defined by, for all non-zero natural integer $n$,
$$w _ { n } = \frac { 3 + \cos ( n ) } { n ^ { 2 } } .$$
Statement 2: The sequence $\left( w _ { n } \right)$ converges to 0.
\item Consider the following algorithm:
$$\begin{aligned}
& U \leftarrow 5 \\
& N \leftarrow 0
\end{aligned}$$
While $U \leqslant 5000$
$$\begin{aligned}
& U \leftarrow 3 \times U - 8 \\
& N \leftarrow N + 1
\end{aligned}$$
End While\\
Statement 3: At the end of execution, the variable $U$ contains the value 5000.
\item We denote $\mathbb { C }$ the set of complex numbers. We consider the equation (E) with unknown $z$ in $\mathbb { C }$
$$( z - \mathrm { i } ) \left( z ^ { 2 } + z \sqrt { 3 } + 1 \right) = 0$$
Statement 4: All solutions of equation (E) have modulus 1.
\item We consider the complex numbers $z _ { n }$ defined by
$$z _ { 0 } = 2 \text { and for all natural integer } n , z _ { n + 1 } = 2 \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 2 } } z _ { n } .$$
We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). For all natural integer $n$, we denote $M _ { n }$ the point with affixe $z _ { n }$.\\
Statement 5: For all natural integer $n$, the point O is the midpoint of the segment $\left[ M _ { n } M _ { n + 2 } \right]$.
\end{enumerate}