\section*{Exercise 5}
5 points

\section*{Candidates who have not followed the specialized course}
Let $k$ be a strictly positive real number.\\
We consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 1 , u _ { 1 } = k$ and, for all natural integer $n$ by:

$$u _ { n + 2 } = \frac { u _ { n + 1 } ^ { 2 } } { k u _ { n } }$$

It is admitted that all terms of the sequence ( $u _ { n }$ ) exist and are strictly positive.

\begin{enumerate}
  \item Express $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$ as functions of $k$.
  \item Using a spreadsheet, the first terms of the sequence ( $u _ { n }$ ) were calculated for two values of $k$. The value of the real number $k$ is entered in cell E 2 .
\end{enumerate}

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E &  & A & B & C & D & E \\
\hline
1 & $n$ & $u ( n )$ &  &  &  & 1 & $n$ & $u ( n )$ &  &  &  \\
\hline
2 & 0 & 1 &  & $k =$ & 2.7182818 & 2 & 0 & 1 &  & $k =$ & 0.9 \\
\hline
3 & 1 & 2.7182818 &  &  &  & 3 & 1 & 0.9 &  &  &  \\
\hline
4 & 2 & 2.7182818 &  &  &  & 4 & 2 & 0.9 &  &  &  \\
\hline
5 & 3 & 1 &  &  &  & 5 & 3 & 1 &  &  &  \\
\hline
6 & 4 & 0.1353353 &  &  &  & 6 & 4 & 1.2345679 &  &  &  \\
\hline
7 & 5 & 0.0067319 &  &  &  & 7 & 5 & 1.6935088 &  &  &  \\
\hline
8 & 6 & 0.000 1234 &  &  &  & 8 & 6 & 2.581 1748 &  &  &  \\
\hline
9 & 7 & $8.315 \mathrm { E } - 07$ &  &  &  & 9 & 7 & 4.3712422 &  &  &  \\
\hline
10 & 8 & $2.061 \mathrm { E } - 09$ &  &  &  & 10 & 8 & 8.2252633 &  &  &  \\
\hline
\end{tabular}
\end{center}