Question

Let \( x_1 = 2 \) and \( x_{n+1} = 2 + \frac{1}{2x_n} \) for all \( n \in \mathbb{N} \). Then, which one of the following is TRUE?

• A. \( x_{n+1} \geq \frac{4}{x_n} \) for all \( n \in \mathbb{N} \), and \( (x_n) \) is a Cauchy sequence
• B. \( x_{n+1} = \frac{4}{x_n} \) for some \( n \in \mathbb{N} \), and \( (x_n) \) is a Cauchy sequence
• C. \( x_{n+1} = \frac{4}{x_n} \) for all \( n \in \mathbb{N} \), and \( (x_n) \) is NOT a Cauchy sequence
• D. \( x_{n+1} \leq \frac{4}{x_n} \) for some \( n \in \mathbb{N} \), and \( (x_n) \) is NOT a Cauchy sequence

Hint:

For sequences defined by recurrence relations, check the convergence behavior to determine if the sequence is Cauchy. A Cauchy sequence is one that converges as \( n \to \infty \).

Answer 1

The Correct Option is A

Step 1: Analyze the recurrence.
We are given the recurrence relation: \[ x_{n+1} = 2 + \frac{1}{2x_n}. \] We observe that this recurrence ensures that \( x_n \) stays bounded and converges. 
Step 2: Check if the sequence is Cauchy.
A sequence is Cauchy if the terms get arbitrarily close as \( n \) increases. In this case, since the recurrence leads to a bounded and convergent sequence, \( (x_n) \) is a Cauchy sequence. 
Final Answer: \[ \boxed{x_{n+1} \geq \frac{4}{x_n} \text{ for all } n \in \mathbb{N}, \text{ and } (x_n) \text{ is a Cauchy sequence.}} \]