\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 $\left( v _ { n } \right)$ 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 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 ( $u _ { n }$ ). 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 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 suite.

\begin{verbatim}
def suite(n):
    U=30
    W=45
    for i in range (1,n+1):
        \mathrm { U } = \mathrm { U } / 2 + 1 0
        W=W/2+U/2+7
    return W
\end{verbatim}

\begin{enumerate}
  \setcounter{enumi}{1}
  \item The execution of suite(1) does not return the term $w _ { 1 }$. How should the function suite be modified so that the execution of 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:
\end{enumerate}

$$w _ { n } = 10 n \left( \frac { 1 } { 2 } \right) ^ { n } + 11 \left( \frac { 1 } { 2 } \right) ^ { n } + 34$$

(b) It is admitted that for every natural integer $n \geq 4$, we have: $0 \leq 10 n \left( \frac { 1 } { 2 } \right) ^ { n } \leq \frac { 10 } { n }$.

What can we deduce about the convergence of the sequence $\left( w _ { n } \right)$?