\textbf{Exercise 5 (5 points) — Candidates who have not followed the speciality course}

Let $( u _ { n } )$ be the sequence defined by $u _ { 0 } = 3 , u _ { 1 } = 6$ and, for all natural integer $n$:

$$u _ { n + 2 } = \frac { 5 } { 4 } u _ { n + 1 } - \frac { 1 } { 4 } u _ { n } .$$

The purpose of this exercise is to study the possible limit of the sequence $( u _ { n } )$.

\section*{Part A:}
We wish to calculate the values of the first terms of the sequence $( u _ { n } )$ using a spreadsheet. We have reproduced below part of a spreadsheet, where the values of $u _ { 0 }$ and $u _ { 1 }$ appear.

\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
 & A & B \\
\hline
1 & $n$ & $u _ { n }$ \\
\hline
2 & 0 & 3 \\
\hline
3 & 1 & 6 \\
\hline
4 & 2 & \\
\hline
5 & 3 & \\
\hline
6 & 4 & \\
\hline
7 & 5 & \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item Give a formula which, entered in cell B4, then copied downwards, allows obtaining values of the sequence $( u _ { n } )$ in column B.
  \item Copy and complete the table above. Approximate values to $10 ^ { - 3 }$ of $u _ { n }$ will be given for $n$ ranging from 2 to 5.
  \item What can be conjectured about the convergence of the sequence $( u _ { n } )$?
\end{enumerate}

\section*{Part B: Study of the sequence}
We consider the sequences $( v _ { n } )$ and $( w _ { n } )$ defined for all natural integer $n$ by:

$$v _ { n } = u _ { n + 1 } - \frac { 1 } { 4 } u _ { n } \quad \text { and } \quad w _ { n } = u _ { n } - 7 .$$

\begin{enumerate}
  \item a. Prove that $( v _ { n } )$ is a constant sequence.\\
b. Deduce that, for all natural integer $n , u _ { n + 1 } = \frac { 1 } { 4 } u _ { n } + \frac { 21 } { 4 }$.
  \item a. Using the result of question 1. b., show by induction that, for all natural integer $n , u _ { n } < u _ { n + 1 } < 15$.\\
b. Deduce that the sequence $( u _ { n } )$ is convergent.
  \item a. Prove that $( w _ { n } )$ is a geometric sequence and specify its first term and common ratio.\\
b. Deduce that, for all natural integer $n , u _ { n } = 7 - \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$.\\
c. Calculate the limit of the sequence $( u _ { n } )$.
\end{enumerate}