Let the sequence $(u_n)$ defined by $u_0 = 0$ and, for all $n \in \mathbb{N}$,
$$u_{n+1} = 5u_n - 8n + 6.$$

\begin{enumerate}
  \item Calculate $u_1$ and $u_2$.
  \item Let $n$ be a natural number.\\
Copy and complete the function \texttt{suite\_u} with argument \texttt{n} below, written in Python language, so that it returns the value of $u_n$.
\begin{verbatim}
def suite_u(n) :
    u = ...
    for i in range(1,n+1) :
        u = ...
    return u
\end{verbatim}
  \item a. Prove by induction that, for all $n \in \mathbb{N}$, $u_n \geqslant 2n$.\\
b. Deduce the limit of the sequence $(u_n)$.\\
c. Let $p \in \mathbb{N}^*$. Why can we assert that there exists at least one integer $n_0$ such that, for all natural integers $n$ satisfying $n \geqslant n_0$, $u_n \geqslant 10^p$?
  \item Prove that the sequence $(u_n)$ is increasing.
  \item We consider the sequence $(v_n)$, defined for all $n \in \mathbb{N}$, by $v_n = u_n - 2n + 1$.\\
a. Below the function \texttt{suite\_u} above, we have written the function \texttt{suite\_v} below:
\begin{verbatim}
def suite_v(n):
    L = []
    for i in range(n+1) :
        L.append(suite_u(i) - 2*i + 1)
    return L
\end{verbatim}
The command ``L.append'' allows us to add, in the last position, an element to the list $L$.\\
When we enter \texttt{suite\_v(5)} in the console, we obtain the following display:
$$\begin{aligned}
& \ggg \text{suite\_v}(5) \\
& [1, 5, 25, 125, 625, 3125]
\end{aligned}$$
Conjecture, for all natural integer $n$, the expression of $v_{n+1}$ as a function of $v_n$. Prove this conjecture.\\
b. Deduce, for all natural integer $n$, the explicit form of $u_n$ as a function of $n$.
\end{enumerate}