We consider the sequence of complex numbers $(z_n)$ defined by $z_0 = \sqrt{3} - i$ and for every natural number $n$:

$$z_{n+1} = (1 + \mathrm{i})z_n.$$

Parts $A$ and $B$ can be treated independently.

\section*{Part A}
For every natural number $n$, we set $u_n = |z_n|$.

\begin{enumerate}
  \item Calculate $u_0$.
  \item Prove that $(u_n)$ is a geometric sequence with common ratio $\sqrt{2}$ and first term 2.
  \item For every natural number $n$, express $u_n$ as a function of $n$.
  \item Determine the limit of the sequence $(u_n)$.
  \item Given a positive real number $p$, we wish to determine, using an algorithm, the smallest value of the natural number $n$ such that $u_n > p$.\\
Copy the algorithm below and complete it with the processing and output instructions, so as to display the sought value of the integer $n$.
\begin{verbatim}
Variables : u is a real number
        p is a real number
        n is an integer
Initialization : Assign to n the value 0
    Assign to u the value 2
Input : Request the value of p
Processing :
Output :
\end{verbatim}
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item Determine the algebraic form of $z_1$.
  \item Determine the exponential form of $z_0$ and of $1 + \mathrm{i}$.\\
Deduce the exponential form of $z_1$.
  \item Deduce from the previous questions the exact value of $\cos\left(\frac{\pi}{12}\right)$.
\end{enumerate}