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.

\begin{enumerate}
  \item 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.
  \item 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.
  \item 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.
\end{enumerate}