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.
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.
For which values of $n$ is the number $z _ { n }$ real?
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.
\begin{enumerate}
\item 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.
\item For which values of $n$ is the number $z _ { n }$ real?
\end{enumerate}