\section*{Part A}
Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 30$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{2} u_{n} + 10$.\\
Let $(v_{n})$ be the sequence defined for every natural integer $n$ by $v_{n} = u_{n} - 20$.
\begin{enumerate}
  \item Calculate the exact values of $u_{1}$ and $u_{2}$.
  \item Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{1}{2}$.
  \item Express $v_{n}$ as a function of $n$ for every natural integer $n$.
  \item Deduce that, for every natural integer $n$, $u_{n} = 20 + 10\left(\frac{1}{2}\right)^{n}$.
  \item Determine the limit of the sequence $\left(u_{n}\right)$. Justify the answer.
\end{enumerate}

\section*{Part B}
Let $(w_{n})$ be the sequence defined for every natural integer $n$ by:
$$\left\{\begin{array}{l} w_{0} = 45 \\ w_{n+1} = \frac{1}{2} w_{n} + \frac{1}{2} u_{n} + 7 \end{array}\right.$$
\begin{enumerate}
  \item Show that $w_{1} = 44.5$.
\end{enumerate}
We wish to write a function \texttt{suite}, in Python language, which returns the value of the term $w_{n}$ for a given value of $n$. We give below a proposal for this function \texttt{suite}.
\begin{verbatim}
def suite(n) :
    U=30
    W=45
    for i in range (1,n+1) :
        U=U/2+10
        W=W/2+U/2+7
    return W
\end{verbatim}
\begin{enumerate}
  \setcounter{enumi}{1}
  \item The execution of \texttt{suite(1)} does not return the term $w_{1}$. How should the function \texttt{suite} be modified so that the execution of \texttt{suite(n)} returns the value of the term $w_{n}$?
  \item a. Show, by induction on $n$, that for every natural integer $n$ we have:
$$w_{n} = 10n\left(\frac{1}{2}\right)^{n} + 11\left(\frac{1}{2}\right)^{n} + 34$$
b. We admit that for every natural integer $n \geqslant 4$, we have: $0 \leqslant 10n\left(\frac{1}{2}\right)^{n} \leqslant \frac{10}{n}$.\\
What can we deduce about the convergence of the sequence $\left(w_{n}\right)$?
\end{enumerate}