We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$,
$$u_{n+1} = \frac{1}{2}\left(u_n + \frac{11}{u_n}\right)$$
We admit that the sequence $(u_n)$ is well defined.

\textbf{Part A - Study of sequence $(u_n)$}

\begin{enumerate}
  \item Give $u_1$ and $u_2$ in the form of irreducible fractions.
  \item We consider the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = \frac{1}{2}\left(x + \frac{11}{x}\right)$$
Prove that function $f$ is increasing on the interval $[\sqrt{11}; +\infty[$.
  \item Prove by induction that for every natural number $n$, we have: $u_n \geqslant u_{n+1} \geqslant \sqrt{11}$.
  \item Deduce that the sequence $(u_n)$ converges to a real limit. We denote this limit by $a$.
  \item After determining and solving an equation of which $a$ is a solution, specify the exact value of $a$.
\end{enumerate}

\textbf{Part B - Geometric application}

For every natural number $n$, we consider a rectangle $R_n$ with area 11 whose width is denoted $\ell_n$ and length $L_n$.\\
The sequence $(L_n)$ is defined by $L_0 = 5$ and, for every natural number $n$,
$$L_{n+1} = \frac{L_n + \ell_n}{2}$$

\begin{enumerate}
  \item a. Explain why $\ell_0 = 2.2$.\\
b. Establish that for every natural number $n$,
$$\ell_n = \frac{11}{L_n}.$$
  \item Verify that the sequence $(L_n)$ corresponds to the sequence $(u_n)$ from Part A.
  \item Show that for every natural number $n$, we have $\ell_n \leqslant \sqrt{11} \leqslant L_n$.
  \item We admit that the sequences $(L_n)$ and $(\ell_n)$ both converge to $\sqrt{11}$. Interpret this result geometrically in the context of Part B.
  \item Here is a script, written in Python language, relating to the sequences studied in this part:
\begin{verbatim}
def heron(n):
    L=5
    l=2.2
    for i in range(n):
        L = (L+l) / 2
        l = 11 / L
    return round(l, 6), round(L, 6)
\end{verbatim}
We recall that the Python function round$(\mathrm{x}, \mathrm{k})$ returns a rounded version of the number x with k decimal places.\\
a. If the user types heron(3) in a Python execution console, what output values does he obtain for $\ell$ and $L$?\\
b. Give an interpretation of these two values.
\end{enumerate}