We consider the sequence $(u_n)$ defined by $u_0 = 3$ and, for every natural integer $n$, by:

$$u_{n+1} = 5u_n - 4n - 3$$

\begin{enumerate}
  \item a. Prove that $u_1 = 12$.\\
b. Determine $u_2$ by detailing the calculation.\\
c. Using a calculator, conjecture the direction of variation and the limit of the sequence $(u_n)$.
  \item a. Prove by induction that, for every natural integer $n$, we have:
$$u_n \geqslant n + 1.$$
b. Deduce the limit of the sequence $(u_n)$.
  \item We consider the sequence $(v_n)$ defined for every natural integer $n$ by:
$$v_n = u_n - n - 1$$
a. Prove that the sequence $(v_n)$ is geometric. Give its common ratio and its first term $v_0$.\\
b. Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$.\\
c. Deduce that for every natural integer $n$:
$$u_n = 2 \times 5^n + n + 1$$
d. Deduce the direction of variation of the sequence $(u_n)$.
  \item We consider the function below, written incompletely in Python language and intended to return the smallest natural integer $n$ such that $u_n \geqslant 10^7$.\\
a. Copy the program and complete the two missing instructions.\\
b. What is the value returned by this function?
\begin{verbatim}
def suite() :
    u = 3
    n = 0
    while...:
        u = ...
        n = n + 1
    return n
\end{verbatim}
\end{enumerate}