The purpose of this exercise is to study sequences of positive terms whose first term $u_0$ is strictly greater than 1 and possessing the following property: for every natural number $n > 0$, the sum of the first $n$ consecutive terms equals the product of the first $n$ consecutive terms. We admit that such a sequence exists and we denote it $(u_n)$. It therefore satisfies three properties:

\begin{itemize}
  \item $u_0 > 1$,
  \item for all $n \geqslant 0, u_n \geqslant 0$,
  \item for all $n > 0, u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.
\end{itemize}

\begin{enumerate}
  \item We choose $u_0 = 3$. Determine $u_1$ and $u_2$.
  \item For every integer $n > 0$, we denote $s_n = u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.

In particular, $s_1 = u_0$.\\
a. Verify that for every integer $n > 0, s_{n+1} = s_n + u_n$ and $s_n > 1$.\\
b. Deduce that for every integer $n > 0$,
$$u_n = \frac{s_n}{s_n - 1}$$
c. Show that for all $n \geqslant 0, u_n > 1$.
  \item Using the algorithm opposite, we want to calculate the term $u_n$ for a given value of $n$.\\
a. Copy and complete the processing part of the algorithm:
\begin{center}
\begin{tabular}{ | r l | }
\hline
Input: & Enter $n$ \\
 & Enter $u$ \\
Processing: & $s$ takes the value $u$ \\
 & For $i$ going from 1 to $n$: \\
 & $u$ takes the value $\ldots$ \\
 & $s$ takes the value $\ldots$ \\
 & End For \\
Output: & Display $u$ \\
\hline
\end{tabular}
\end{center}
b. The table below gives values rounded to the nearest thousandth of $u_n$ for different values of the integer $n$:
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$n$ & 0 & 5 & 10 & 20 & 30 & 40 \\
\hline
$u_n$ & 3 & 1.140 & 1.079 & 1.043 & 1.030 & 1.023 \\
\hline
\end{tabular}
\end{center}
What conjecture can be made about the convergence of the sequence $(u_n)$?
  \item a. Justify that for every integer $n > 0, s_n > n$.\\
b. Deduce the limit of the sequence $(s_n)$ then that of the sequence $(u_n)$.
\end{enumerate}