Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$

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Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$

I noticed that the following identies hold with the help of wolfram alpha and oeis. I'm sure they're well-known, but I'd like to know how they generalize. ...
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55 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 ...
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41 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 ...
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1answer
58 views

Rational aproximations of golden ratio

I read a blogpost that mentions that for golden ratio, the sets of best rational approximations of the first kind and the second kind are the same. Is this true? If so, why? Are there other numbers ...
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Golden Ratio method question

I am reading a link on the Golden Ratio Method from http://mathfaculty.fullerton.edu/mathews/n2003/GoldenRatioSearchMod.html The part where it says If $f(c) \leq f(d)$ and only one new ...
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69 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 ...
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674 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 $$ ...
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65 views

A sequence that converges to the golden ratio

This is elementary, but I found it somewhat surprising. Define $$ a_n = \frac{1+ a_{n-2}}{\sqrt{1+a_{n-1}}} \;,$$ where $a_1$ and $a_2$ are constants. For example, here is a plot for $a_1=5$ and ...
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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 ...
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87 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 ...
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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 ...
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64 views

How does this proof of Fibonacci work

\begin{eqnarray*} F_{i+1}&=&F_{i} + F_{i-1}\\ &=&\frac{\phi^i-\hat{\phi^i}}{\sqrt5}+\frac{\phi^{i-1}-\hat{\phi^{i-1}}}{\sqrt5}\\ ...
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2answers
205 views

How To Determine If A Large Number is Prime?

For a very large number n, how many divisibility tests are required to establish if its prime? I know this has something to do with the Golden Number, but I can't figure out what. I did try searching ...
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40 views

$e_{n+1} = K e_n e_{n-1} $ is $|e_{n+1}| = C|e_n|^{\varphi}$?

if $ e_{n+1} = K e_n e_{n-1} $ ($K$ is a constant, and $e_n$ is a serise), then, $ | e_{n+1} | = C|e_n|^\varphi$($C$ is constant) and $\varphi$ is golden ratio. Is this true? If yes, How can I show ...
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Source for relationsheep 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: ...
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132 views

Representation of integers as powers of the golden ratio

How to prove that any integer $n$ can be represented in the form of $$n= \phi^{z_1}+\phi^{z_2}+\phi^{z_3}+...+\phi^{z_m}$$ For $z_1$, $z_2$... $z_m$ $\in$ $\mathbb Z $ and $\phi =\frac{ \sqrt ...
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Finding relatives of the series $\varphi =\frac{3}{2}+\sum_{k=0}^{\infty}(-1)^{k}\frac{(2k)!}{(k+1)!k!2^{4k+3}}$.

Consider $\varphi=\frac{1+\sqrt{5}}{2}$, the golden ratio. Bellow are series $(3)$ and $(6)$ that represent $\varphi$ $$ \begin{align*} \varphi &=\frac{1}{1}+\sum_{k=0}^{\infty}\cdots&(1)\\ ...
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522 views

Is there an identity that links $\pi$ and $\phi$ (the golden ratio)? [duplicate]

Is there some identity that shows a connection between $\pi$ and the golden ratio, $\phi$?
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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) ...
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3answers
251 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 ...
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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
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Best way to discover the 'golden ratio'

Let's say you live in a world where nobody ever discovered the Golden ratio. What's the most intuitive way to discover this proportion? Wikipedia defined it this way: $$\phi = \frac{a+b}{a} = ...
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117 views

Golden spiral created using golden rectangles vs pentagram

I am trying to create a graphic that shows the golden spiral created using a pentagram and the golden triangles contained therein. I have drawn out the pentagram and golden triangles and the ...
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280 views

On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as ...
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304 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} ...
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53 views

Is there a solution of $(x-1)/x=1$ or $(x+1)/x=1$?

Is there a solution of $(x-1)/x=1$ or $(x+1)/x=1$? Layman is trying to reversing golden ratio. Tell a real solution and complex solution please. Intuition says there should be a real value of ...
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376 views

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: ...
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Hint to prove that $\phi^n + \phi'^n$ is an integer.

I was solving some induction exercises but I found this that I could not solve. Let $n \in \mathbb{N}$, prove that $\phi^n + \phi'^n$ is an integer where $\phi=\frac{1+\sqrt{5}}{2}$ and ...
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181 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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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 ...
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pandigital rational approximations to the golden ratio and the base of the natural logarithm

Steven Stadnicki suggested in a comment that I post the following as a question. The golden ration $\phi$ is given by $$\phi = \frac{1+\sqrt{5}}{2} \approx 1.618033988.$$ A rational approximation is ...
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How to prove that $\frac{1}{n+1} = {\sqrt{n\over n+1}}\implies n = \Phi$?

Consider: $$\frac{1}{n+1} = {\sqrt{n\over n+1}}$$ How could one prove that $n$ is of such form that: $$\frac{1}{n+1} = {\sqrt{n\over n+1}}\implies n = {\sqrt{5\,}-1 \over 2} \implies n = \Phi$$ ...
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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 ...
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Golden ratio rectangles

I'm designing a layout and I would like to use four golden ratio rectangles. The total width of the layout is 960px. How do I find the height (x)? Below is a diagram of the layout.
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Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
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What real number is exactly one less than its cube?

And does it have any of the special properties that the golden ratio (one less than its square) has?
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116 views

Can rainbows be considered to have mathematical patterns?

Of course it is evident that there is physics and mathematics involved in a rainbow, but my question is (as the title suggests) are there patterns to be found in rainbows? By patterns I mean ...
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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 ...
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269 views

Continued Fraction [1,1,1,…]

If the continued fractional representation of an irrational number $\alpha$ is given by [1,1,1,...], I can compute that $\alpha = \frac{1+\sqrt{5}}{2}$ by solving the equation $\alpha = 1+ ...
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understanding the Golden ratio intuitively

i am very interested in Golden Ratio and its value. the Golden Ratio itself is not hard thing to visualize and understand in 5 minutes. But i am trying to reach the historical, logical reasons of ...
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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, ...
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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 ...
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Computing $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$

I'm trying to find $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$ mod $m$. $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\varphi^3 = 2 + \sqrt{5}$. But honestly I'm not even sure where to start. ...
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Prove that the sequence $x_{n+1} = {\frac{x_n+1}{x_n+2}}, x_0 = 1$ converges to $\frac{1}{\varphi}$

I've established that $(x_n)$ is monotonically decreasing, but don't know if I should attempt to show that $\frac{1}{\varphi}$ is the infimum of $(x_n)$ or use another method.
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166 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 ...
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Is the golden ratio overrated?

While reading many sources and watching many video's about nature and physics I wonder if the golden ratio is really that often occurring in nature and physics ? If I look on the Wikipedia pages I see ...
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Does this number have an expression as square root or log of something

If this ends up being a ridiculous question I will delete it. Forgive me if this is ridiculous but this number has me stumped. $$1.52360679774998$$ The continued fraction calculator gives $1, 1, 1, ...
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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 ...
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Is it possible to extend phi to higher order polynomials?

Below, $x=\phi$ when $n=2$: $$x^n-\sum_{i=1}^{n}x^{n-i}=0$$ ($\phi$ being the golden ratio) Is there a way to express $x$ in terms of $\phi$ for $n>2$?
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Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?

I have been looking at concise ways to represent irrational numbers using only integers. I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...