Question

Let \( x_1 = \frac{5}{2} \) and for \( n \in \mathbb{N} \), define \[ x_{n+1} = \frac{1}{5} \left( x_n^2 + 6 \right). \] Then, which one of the following is TRUE?

• A. \( (x_n) \) is an increasing sequence, and \( (x_n) \) is NOT a bounded sequence
• B. \( (x_n) \) is NOT an increasing sequence, and \( (x_n) \) is NOT a bounded sequence
• C. \( (x_n) \) is NOT a decreasing sequence, and \( (x_n) \) is a bounded sequence
• D. \( (x_n) \) is a decreasing sequence, and \( (x_n) \) is a bounded sequence

Hint:

For recursive sequences, find the fixed points to check if the sequence is bounded. Use monotonicity tests to determine if the sequence is increasing or decreasing.

Answer 1

The Correct Option is D

Step 1: Analyze the recurrence relation.
The recurrence relation \( x_{n+1} = \frac{1}{5} \left( x_n^2 + 6 \right) \) indicates that the sequence \( x_n \) is increasing if \( x_n \) is large enough, but it's not guaranteed to be strictly increasing for all \( n \). 
Step 2: Show that the sequence is bounded.
To show that \( x_n \) is bounded, note that if \( x_n \) is large, the sequence will converge towards a fixed point \( L \), where \( L = \frac{1}{5} \left( L^2 + 6 \right) \). Solving this, we find that the sequence is bounded above and below. 
Final Answer: \[ \boxed{(x_n) \text{ is a decreasing sequence, and } (x_n) \text{ is a bounded sequence.}} \]