Question

Let \( a_n = \frac{b_{n+1}}{b_n} \), where \( b_1 = 1 \), \( b_2 = 1 \), and \( b_{n+2} = b_n + b_{n+1} \), \( n \in \mathbb{N} \). Then \( \lim_{n \to \infty} a_n \) is

• A. \( \frac{1 - \sqrt{5}}{2} \)
• B. \( \frac{1 - \sqrt{3}}{2} \)
• C. \( \frac{1 + \sqrt{3}}{2} \)
• D. \( \frac{1 + \sqrt{5}}{2} \)

Hint:

The limit of the ratio of consecutive Fibonacci numbers is the golden ratio, \( \frac{1 + \sqrt{5}}{2} \).

Answer 1

The Correct Option is D

Step 1: Understanding the sequence.
The recurrence relation \( b_{n+2} = b_n + b_{n+1} \) defines the Fibonacci sequence. We are interested in the limit of the ratio \( a_n = \frac{b_{n+1}}{b_n} \) as \( n \to \infty \). This ratio converges to the golden ratio.

Step 2: Analyzing the options.
(A) \( \frac{1 - \sqrt{5}}{2} \): This is the negative reciprocal of the golden ratio.
(B) \( \frac{1 - \sqrt{3}}{2} \): This is incorrect and does not relate to the Fibonacci sequence.
(C) \( \frac{1 + \sqrt{3}}{2} \): This is not the correct value for the limit of the ratio of Fibonacci numbers.
(D) \( \frac{1 + \sqrt{5}}{2} \): Correct — This is the golden ratio, which is the limit of the ratio of consecutive Fibonacci numbers.

Step 3: Conclusion.
The correct answer is (D) \( \frac{1 + \sqrt{5}}{2} \).