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: $$z_{n+1} = 1 - \frac{1}{z_{n}}$$
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.
Determine $z_{2016}$ in the case where $z_{0} = 1 + \mathrm{i}$.
Are there values of $z_{0}$ such that $z_{0} = z_{1}$? What can we say about the sequence $(z_{n})$ in this case?
\section*{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:
$$z_{n+1} = 1 - \frac{1}{z_{n}}$$
\begin{enumerate}
\item 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.
\item Determine $z_{2016}$ in the case where $z_{0} = 1 + \mathrm{i}$.
\item Are there values of $z_{0}$ such that $z_{0} = z_{1}$? What can we say about the sequence $(z_{n})$ in this case?
\end{enumerate}