The exercise involves a sequence $(U_n)$, which is defined recursively by the relation:

$U_{n+1} = \frac{5U_n - 3}{U_n + 1}$ with the initial condition that $U_0 = c$.

Let's break it down step by step as requested in the exercise:

1) Calculer $U_1$ et $U_2$

We need to compute the first two terms of the sequence, $U_1$ and $U_2$, given $U_0 = c$. To find these, we can substitute $U_0 = c$ into the recursive relation.

$U_1 = \frac{5U_0 - 3}{U_0 + 1} = \frac{5c - 3}{c + 1}$ $U_2 = \frac{5U_1 - 3}{U_1 + 1} = \frac{5\left( \frac{5c - 3}{c + 1} \right) - 3}{\left( \frac{5c - 3}{c + 1} \right) + 1}$

2) Mq par récurrence

We are asked to prove that $(U_n)$ is bounded such that:

$\forall n \in \mathbb{N}, \quad 1 < U_n < 3$

To do this, we would typically use induction to show that the sequence stays between these bounds. This would involve showing that if $1 < U_n < 3$, then $1 < U_{n+1} < 3$ as well.

3) a) Mq

This part likely asks to express the recurrence relation and manipulate it algebraically. The equation $U_{n+1} - U_n = - \frac{(U_n - 1)(U_n - 3)}{U_n + 1}$ is given, so we could simplify or analyze it further.

3) b) Monotonicity

To determine the monotonicity of the sequence, we need to analyze whether the sequence is increasing or decreasing. This would involve finding the sign of $U_{n+1} - U_n$ and determining if it is positive or negative.

4) a) Mq (suite géométrique)

In this step, we introduce a new sequence $V_n = U_n - 3$. This will help simplify the recurrence. We can then examine the new recurrence and show that $(V_n)$ is a geometric sequence.

4) b) Expression for $V_n$ and $U_n$

Next, we express $V_n$ and $U_n$ in terms of $n$.

4) c) Mq

Finally, we need to express the sum $V_2 + V_3 + \cdots + V_n$ in a closed form. The given formula suggests that the sum can be written as:

$\frac{1}{2} \left( 1 - \frac{1}{2^{n-1}} \right)$

Would you like me to elaborate on any of these steps or provide a detailed solution for one of them?