https://r2.erweima.ai/mathbot/image/1727182866322fhd8oy2s.jpg

The exercise involves analyzing two sequences \( (v_n) \) and \( (w_n) \) defined in relation to another sequence \( u_n \). Let's break it down and work through the questions:

Given:
- \( u_0 = 0 \), \( u_1 = 1 \)
- For all \( n \), \( v_n = u_{n+1} - u_n \) and \( w_n = \frac{u_{n+2} + 2u_{n+1} + u_n}{3} \)

### Questions:

1. **Prove that the sequence \( v_n \) is a geometric sequence.**
   To do this, we need to show that the ratio between consecutive terms of \( v_n \) is constant, i.e., \( \frac{v_{n+1}}{v_n} = r \) for some constant \( r \).

2. **Prove that the sequence \( w_n \) is a constant sequence.**
   This involves showing that \( w_n = w_{n+1} \) for all \( n \), implying that the sequence does not change and thus is constant.

3. **Deduce the expression of \( u_n \) as a function of \( n \).**
   This requires finding an explicit formula for \( u_n \) based on the properties derived from the previous steps.

4. **Determine the limit of \( u_n \).**
   Based on the expression derived in question 3, we calculate the limit \( \lim_{n \to \infty} u_n \).

Let me know if you'd like me to proceed with solving these questions!

Exercice n°3: On considère la suite u définie par u_0=0 et u_1=1 et pour tout entier n, v_n=u_{n+1}-u_n et w_n=(u_{n+2}+2u_{n+1}+u_n)/3. 1) Démontrer que la suite (v_n) est une suite géométrique. 2) Démontrer que la suite (w_n) est une suite constante. 3) En déduire l'expression de u_n en fonction de n. 4) Déterminer la limite de u_n.

Solve a recursive sequence problem involving a geometric sequence (v_n) and a constant sequence (w_n). Learn to prove these properties and deduce the expression of u_n while determining the limit of u_n as n approaches infinity.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation