4
votes
0answers
76 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
1
vote
1answer
60 views

Proof that golden angle successively divides the largest gap by the golden ratio?

The golden angle divides the circumference of a circle by the golden ratio. "If radial spokes are placed successively into the circle, each spaced by a golden angle increment, then each additional ...
0
votes
1answer
42 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
3
votes
3answers
80 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
6
votes
4answers
692 views

Fibonacci numbers and golden ratio

Let $\Phi$ be the golden ratio and $F_n$ be the usual Fibonacci numbers. How can I derive the following formula? $$ \Phi = \lim_{n\rightarrow \infty} \sqrt[n]{F_n} $$ I know the usual relation $$ ...
1
vote
5answers
85 views

Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$ [duplicate]

We define the $n^{th}$ Fibonacci number as $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. Consider $$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}. $$ I wrote a script and found that this ...
0
votes
2answers
88 views

Prove that a Fibonacci number is greater than $ φ^n$

How can I prove the following: If $f_n$ is a number of the Fibonacci sequence and φ= $\frac{1+\sqrt{5}}2$, then $f_n > φ^n$ for every $n >2$? I have tried using induction but I can't ...
4
votes
2answers
277 views

Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate]

Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists. How can we prove ...
2
votes
1answer
62 views

Source for relationship between $d$-ary Fibonacci numbers and generalized golden ratio?

I am not a mathematician (but a computer scientist) and stumbled across the following in the analysis of an algorithm (Berthold Vöcking: How Asymmetry Helps Load Balancing). The author gives Knuth: ...
-1
votes
3answers
118 views

Fibonacci Sequence or Golden Ratio?

Using the polar coordinate system, $r$ increases directly with $\theta$. In other words, $r=k\theta$. Which of the following shapes is constructed? A) Fibonacci Sequence B) Golden Ratio C) ...
2
votes
3answers
303 views

Limit of the ratio of consecutive Fibonacci numbers [duplicate]

I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...
2
votes
3answers
127 views

Golden ratio / Fibonacci which branch of math?

Friends, The Golden ratio / Fibonacci sequence are studied under which branch of math? Can you recommend some good textbooks on the subject? Thanks
0
votes
1answer
336 views

Induction on Fibonacci Sequence and the Golden Ratio

I'm trying to prove $f_n \leq \left(\frac{1 + \sqrt{5}}{2}\right)^{n-1}$ with induction, and I'm stuck in the induction step. Basis: n = 2 $f_{2} \leq \left(\frac{1 + \sqrt{5}}{2}\right)^{2-1} ...
2
votes
0answers
86 views

Why is the reciprocal of the second Fibonacci number negative?

The second Fibonacci number is 1, so it's reciprocal should be 1, right? Why is it that I get $-1$ when I plug in $2$ for n in the reciprocal of Binet's equation ...
0
votes
2answers
138 views

intuition for the closed form of the fibonacci sequence

I'm trying to picture this closed form from Wikipedia visually: The idea is, if you take $\phi^n / \sqrt{5}$ and round it to the nearest integer, you'll get the $n$th Fibonacci number. I see ...
2
votes
2answers
354 views

How can I find an inverse to the Binet formula?

I'm already aware of the Binet formula $F_n = \frac{\varphi^n + \frac{1}{\varphi^n}}{\sqrt{5}}$. I'm attempting to find the inverse of that formula so I can find the position in the sequence of ...
2
votes
2answers
59 views

Non-isomorphic combinatorial classes with growth rate equal to the golden ratio?

In a couple of weeks, I'm giving a talk about the growth rates of some combinatorial classes. In the introduction, I thought I'd present the class of tilings of a $n\!\times\!2$ strip with dominos, ...
4
votes
3answers
239 views

Why doesn't this induction “proof ”show $f_n = (\phi)^n + (1-\phi)^n$?

Here, $\phi$ is the golden ratio and $f_n$ is the $n^{th}$ Fibonacci number. The formula I'm using is actually the closed form of the Lucas numbers. Let $n = 1$. Then $f_n = 1$ and $\phi + 1 - \phi ...
1
vote
1answer
172 views

Need formula for sequence related to Lucas/Fibonacci numbers

I am trying to get a formula for the nth term of the following sequence: 2, 14, 22, 58, 266, 398, 1042, 4774, 7142, 18698, 85666, 128158, 335522,... It's not in OEIS and as far as I can tell ...
0
votes
0answers
85 views

Finding the kth n-anacci number

At this wolfram link the formula for the kth n-anacci number is given: http://mathworld.wolfram.com/Fibonaccin-StepNumber.html#eqn8 (Eq. 4) Not sure if I understand correctly. If I want the fifth ...
0
votes
2answers
237 views

Solving Fibonaccis Term Using Golden Ratio Convergance

While solving this problem, I discovered that there is a relationship between the Fibonacci sequence and the golden ratio. After I got the correct answer via brute force, I discovered this ...
4
votes
1answer
168 views

Golden parallelepiped

Define a golden parallelepiped as a $d$-dimensional box with side lengths $(1, \phi, \phi^2, \ldots, \phi^{d-1})$, where $\phi$ is the golden ratio: ...
6
votes
3answers
411 views

How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$

How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$ where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio?