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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?

Soit ( U_n ) la suite numérique définie par U_∞ = c et pour tout n ∈ ℕ, U_{n+1} = (5U_n - 3) / (U_n + 1). Calculer : U_1 et U_2. Mq par récurrence (∀n ∈ ℕ, 1 < U_n < 3). Déterminer la monotonie de la suite (U_n). Pour tout n ∈ ℕ, on pose V_n = U_n - 3, U_n - 1. Mq (U_n) est une suite géométrique dont on précisera la raison et le premier terme.

This exercise focuses on calculating the first terms of a recursive sequence, proving the boundedness using induction, determining the monotonicity, and expressing a sequence in terms of a geometric progression. It involves advanced concepts in sequences and series. Suitable for high school students in Grades 11-12.

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