Math Problem Statement

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.

Solution

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

Un+1=5Un3Un+1U_{n+1} = \frac{5U_n - 3}{U_n + 1} with the initial condition that U0=cU_0 = c.

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

1) Calculer U1U_1 et U2U_2

We need to compute the first two terms of the sequence, U1U_1 and U2U_2, given U0=cU_0 = c. To find these, we can substitute U0=cU_0 = c into the recursive relation.

U1=5U03U0+1=5c3c+1U_1 = \frac{5U_0 - 3}{U_0 + 1} = \frac{5c - 3}{c + 1} U2=5U13U1+1=5(5c3c+1)3(5c3c+1)+1U_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 (Un)(U_n) is bounded such that:

nN,1<Un<3\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<Un<31 < U_n < 3, then 1<Un+1<31 < U_{n+1} < 3 as well.

3) a) Mq

This part likely asks to express the recurrence relation and manipulate it algebraically. The equation Un+1Un=(Un1)(Un3)Un+1U_{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 Un+1UnU_{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 Vn=Un3V_n = U_n - 3. This will help simplify the recurrence. We can then examine the new recurrence and show that (Vn)(V_n) is a geometric sequence.

4) b) Expression for VnV_n and UnU_n

Next, we express VnV_n and UnU_n in terms of nn.

4) c) Mq

Finally, we need to express the sum V2+V3++VnV_2 + V_3 + \cdots + V_n in a closed form. The given formula suggests that the sum can be written as:

12(112n1)\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?

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Math Problem Analysis

Mathematical Concepts

Recursive Sequences
Inductive Proof
Geometric Sequences
Monotonicity

Formulas

U_{n+1} = (5U_n - 3) / (U_n + 1)
V_n = U_n - 3
Sum of geometric series formula: S_n = a(1 - r^n) / (1 - r)

Theorems

Mathematical Induction
Monotonicity of Sequences
Geometric Series Sum Formula

Suitable Grade Level

Grades 11-12