Question

1. the fibonacci sequence starts with 1 (or 0) and is followed by 1. from there, each term is found by adding the two previous terms, as shown below: 1, 1, 2, 3, 5, 8, 13, ... a) does this sequence have a limit? explain. b) if you move through the sequence and divide each number by the previous number, the results approach a very famous number called the golden ratio. determine this limit, rounded to five decimal places. c) instead of dividing each term by the previous term, if you divide each term by the next term, the results approach another value related to the golden ratio. determine this value, rounded to five decimal places. d) do you notice anything interesting when you compare the values found in parts (b) and (c)? explain. can you see how the fibonacci sequence and the golden ratio appear in this diagram of the golden spiral?

Explanation:

Step1: Analyze limit of Fibonacci sequence

The Fibonacci sequence \(F_n=F_{n - 1}+F_{n - 2}\), with \(F_1 = 1,F_2=1\). As \(n\to\infty\), the terms of the sequence grow without bound. So \(\lim_{n
ightarrow\infty}F_n=\infty\), it does not have a finite - limit.

Step2: Find the limit of ratio of consecutive terms

Let \(\lim_{n
ightarrow\infty}\frac{F_{n+1}}{F_n}=L\). Since \(F_{n + 1}=F_n+F_{n - 1}\), then \(\lim_{n
ightarrow\infty}\frac{F_{n+1}}{F_n}=\lim_{n
ightarrow\infty}(1 + \frac{F_{n - 1}}{F_n})\). Because \(\lim_{n
ightarrow\infty}\frac{F_{n+1}}{F_n}=\lim_{n
ightarrow\infty}\frac{F_n}{F_{n - 1}}=L\), we have \(L = 1+\frac{1}{L}\). Rearranging gives \(L^2 - L - 1=0\). Using the quadratic formula \(L=\frac{1\pm\sqrt{1 + 4}}{2}\). Since \(L>0\), \(L=\frac{1+\sqrt{5}}{2}\approx1.61803\).

Step3: Find the limit of ratio of non - consecutive terms

Let's find \(\lim_{n
ightarrow\infty}\frac{F_n}{F_{n + 1}}\). Since \(\lim_{n
ightarrow\infty}\frac{F_{n+1}}{F_n}=L=\frac{1 + \sqrt{5}}{2}\), then \(\lim_{n
ightarrow\infty}\frac{F_n}{F_{n + 1}}=\frac{1}{L}=\frac{2}{1+\sqrt{5}}=\frac{\sqrt{5}-1}{2}\approx0.61803\).

Step4: Compare the two limits

The value in part (b) is the golden ratio \(\varphi=\frac{1+\sqrt{5}}{2}\), and the value in part (c) is \(\frac{1}{\varphi}=\frac{\sqrt{5}-1}{2}\). They are reciprocals of each other.

Answer:

a) No. The terms of the Fibonacci sequence grow without bound as \(n\to\infty\).
b) \(1.61803\)
c) \(0.61803\)
d) The value in part (b) is the golden ratio \(\varphi\), and the value in part (c) is \(\frac{1}{\varphi}\). They are reciprocals of each other.