Question

Where Fn is the fibonacci sequence find the lim (Fn+1)/F(n)

Answer 1

The limit of the ratio of consecutive Fibonacci numbers as n approaches infinity is known to converge to the golden ratio, which is approximately 1.618033988749895.

Step-by-step explanation:

The student is asking about the limit of the ratio of consecutive Fibonacci numbers as n approaches infinity. The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. The limit of the ratio of consecutive Fibonacci numbers (Fn+1/Fn) as n approaches infinity is known to converge to the golden ratio, φ (phi), which is approximately 1.618033988749895.

Here is a step-by-step explanation:

1. Start with the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, and so forth, where each term is the sum of the two previous terms.
2. Form a sequence of ratios of consecutive terms: 1/1, 2/1, 3/2, 5/3, 8/5, and so on.
3. Notice that as n increases, the ratios approach the same number, which is the golden ratio, φ (phi).

It can be proven that this limit equals the golden ratio through algebra by setting up a quadratic equation that these ratios satisfy and solving for the ratio as n approaches infinity.