Question

Let \( \{a_n\} \) be a sequence defined as follows:
\[ a_1 = 1 \quad \text{and} \quad a_{n+1} = \frac{7a_n + 11}{21}, \quad n \in \mathbb{N}. \] Which of the following statements is TRUE?

• A. \( \{a_n\} \) is an increasing sequence which diverges
• B. \( \{a_n\} \) is an increasing sequence with \( \lim_{n \to \infty} a_n = \frac{11}{14} \)
• C. \( \{a_n\} \) is a decreasing sequence which diverges
• D. \( \{a_n\} \) is a decreasing sequence with \( \lim_{n \to \infty} a_n = \frac{11}{14} \)

Hint:

When working with recurrence relations, set \( a_n = a_{n+1} = L \) and solve for \( L \) to find the limit of the sequence.

Answer 1

The Correct Option is D

Step 1: Finding the limit of the recurrence relation.
We start by solving the recurrence relation for the limit. Let \( L = \lim_{n \to \infty} a_n \). Assuming the sequence converges, we have: \[ L = \frac{7L + 11}{21} \] Multiplying both sides by 21 and solving for \( L \): \[ 21L = 7L + 11 \quad \Rightarrow \quad 14L = 11 \quad \Rightarrow \quad L = \frac{11}{14} \]
Step 2: Determining whether the sequence is increasing or decreasing.
To analyze whether the sequence is increasing or decreasing, observe that the recurrence relation shows that each term \( a_{n+1} \) is smaller than the previous term, so the sequence is decreasing.
Step 3: Conclusion.
The sequence is decreasing and converges to \( \frac{11}{14} \), so the correct answer is (D).