Consider the sequences $\left(v_n\right)$ and $\left(w_n\right)$ defined for every natural integer $n$ by:

$$\left\{ \begin{array}{ll} v_0 &= \ln(4) \\ v_{n+1} &= \ln\left(-1 + 2\mathrm{e}^{v_n}\right) \end{array} \quad \text{and} \quad w_n = -1 + \mathrm{e}^{v_n} \right.$$

We admit that the sequence $\left(v_n\right)$ is well defined and strictly positive.

\begin{enumerate}
  \item Give the exact values of $v_1$ and $w_0$.
  \item a. Among the three formulas below, choose the formula which, entered in cell B3 then copied downward, will allow you to obtain the values of the sequence $(v_n)$ in column B.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Formula 1 & $\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{B}2)\right)$ \\
\hline
Formula 2 & $=\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{B}2)\right)$ \\
\hline
Formula 3 & $=\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{A}2)\right)$ \\
\hline
\end{tabular}
\end{center}
b. Conjecture the direction of variation of the sequence $\left(v_n\right)$.\\
c. Using a proof by induction, validate your conjecture concerning the direction of variation of the sequence $(v_n)$.
  \item a. Prove that the sequence $(w_n)$ is geometric.\\
b. Deduce that for every natural integer $n$, $v_n = \ln\left(1 + 3 \times 2^n\right)$.\\
c. Determine the limit of the sequence $\left(v_n\right)$.
  \item Justify that the following algorithm written in Python language returns a result regardless of the choice of the value of the number S.
\begin{verbatim}
from math import*
def seuil(S):
    V=ln(4)
    n=0
    while V < S :
        n=n+1
        V=ln(2*exp(V)-1)
    return(n)
\end{verbatim}
\end{enumerate}