\textbf{Exercise 4 (Candidates who have not followed the mathematics specialization course)}

Consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = \frac { 1 } { 2 }$ and such that for every natural integer $n$,

$$u _ { n + 1 } = \frac { 3 u _ { n } } { 1 + 2 u _ { n } }$$

\begin{enumerate}
  \item a. Calculate $u _ { 1 }$ and $u _ { 2 }$.\\
b. Prove, by induction, that for every natural integer $n , 0 < u _ { n }$.
  \item We admit that for every natural integer $n , u _ { n } < 1$.\\
a. Prove that the sequence $\left( u _ { n } \right)$ is increasing.\\
b. Prove that the sequence $( u _ { n } )$ converges.
  \item Let $\left( v _ { n } \right)$ be the sequence defined, for every natural integer $n$, by $v _ { n } = \frac { u _ { n } } { 1 - u _ { n } }$.\\
a. Show that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio 3.\\
b. Express for every natural integer $n , v _ { n }$ as a function of $n$.\\
c. Deduce that, for every natural integer $n , u _ { n } = \frac { 3 ^ { n } } { 3 ^ { n } + 1 }$.\\
d. Determine the limit of the sequence $\left( u _ { n } \right)$.
\end{enumerate}