We define, for every natural integer $n$, the complex numbers $z$ by:

$$\begin{cases} z_{0} & = 16 \\ z_{n+1} & = \frac{1 + \mathrm{i}}{2} z_{n}, \text{ for every natural integer } n. \end{cases}$$

We denote $r_{n}$ the modulus of the complex number $z_{n}$: $r_{n} = |z_{n}|$. In the plane equipped with a direct orthonormal coordinate system with origin O, we consider the points $A_{n}$ with affixes $z_{n}$.

\begin{enumerate}
  \item a) Calculate $z_{1}, z_{2}$ and $z_{3}$.\\
b) Plot the points $A_{1}$ and $A_{2}$ on the graph given in the appendix, to be returned with your answer sheet.\\
c) Write the complex number $\frac{1 + \mathrm{i}}{2}$ in trigonometric form.\\
d) Prove that the triangle $\mathrm{OA}_{0}A_{1}$ is isosceles right-angled at $A_{1}$.
  \item Prove that the sequence $(r_{n})$ is geometric, with common ratio $\frac{\sqrt{2}}{2}$.
\end{enumerate}

Is the sequence $(r_{n})$ convergent?\\
Interpret the previous result geometrically.\\
We denote $L_{n}$ the length of the broken line connecting point $A_{0}$ to point $A_{n}$ passing successively through points $A_{1}, A_{2}, A_{3}$, etc.\\
Thus $L_{n} = \sum_{i=0}^{n-1} A_{i}A_{i+1} = A_{0}A_{1} + A_{1}A_{2} + \ldots + A_{n-1}A_{n}$.\\
3. a) Prove that for every natural integer $n$: $A_{n}A_{n+1} = r_{n+1}$.\\
b) Give an expression for $L_{n}$ as a function of $n$.\\
c) Determine the possible limit of the sequence $(L_{n})$.