Questions tagged [golden-ratio]
Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$
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Just as Fibonacci addition approaches phi with a(n-1)*phi = a(n), Fibonacci multiplication approaches phi with a(n-1)^phi = a(n). Can we extend?
Fibonacci addition converges on the golden ratio between consecutive values, i.e. a(n)/a(n-1) = phi. Example: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... 55/34 rounds to 1.618.
I realized today that Fibonacci ...
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Is there a clever way to show that $\sin (\phi+1)<\frac12$?
I noticed that $\phi+1\approx 1.000015\left(\frac{5\pi}{6}\right)$, where $\phi=\frac{1+\sqrt5}{2}$, the golden ratio.
So I wonder, is there a clever way to show that $\sin (\phi+1)<\frac12$, ...
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(answered) found the golden ratio in something out of pure luck and want to know if there is any reason?
As a background I know nothing about how the golden ratio is used in actual mathematics or any formulae and such (only seen it used in a few examples I've seen online).
But then while messing on the ...
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A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$
I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
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A conjecture for the golden ration via continued fraction $\phi=\frac{1+\sqrt{5}}{2}$
Playing with my own question Got a factored version of the Taylor's series? I found that :
Define :
$$f(x)=\frac{1}{\sqrt{x-2/x}}$$
Then it seems we have :
$$\phi=f(f(\cdots f(\phi)\cdots))$$
...
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Why is the continued fraction [1,1,1,1,1...] equal to ϕ?
I wanted to know what the continued fraction $1 + \frac 1{1 + \frac 1{1+ \frac 1{1+\frac 1{1+ \frac 1{1 + \ldots}}}}}$ would euqal to,
so I chose $x=1+\frac1{x}$ because you can put the euqation in ...
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Explain this proof of Binet's formula?
I am confused about the proof offered here: https://sicp-solutions.net/post/sicp-solution-exercise-1-13/.
The proof starts at Fib(n) = Fib(n - 1) + Fib(n - 2), which is true by definition. Then it ...
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Is there a graph that satisfies the golden ratio polynomial?
Is there a graph $G$ containing a bridge-edge $e$, such that if you delete the edge $e$, the resulting graph $G-e$ is isomorphic to $G\times G$?
Such a graph, if it exists, would be a graph analogue ...
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Is my reasoning using limits correct for why 𝜙=1+1/𝜙?
let $$\lim_{n \to \infty}\frac{f_{n+1}}{f_n} = \phi$$ by definition, where $$f_n$$ is the nth fibonacci number
then $$\implies \lim_{n\to \infty}1+\frac{f_{n-1}}{f_n}= \phi $$
$$\implies 1+\lim_{n\to \...
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Which $x \in \mathbb R$ satisfy a property related to sequences
This is a follow up to If $\sum\limits_{n=1}^{\infty}e_n x^n = 0$ always implies $\sum\limits_{n=1}^{\infty}e_n a_n = 0$, then $(a_n) = (x^n)$?.
For $x \in \mathbb R$ we define the property $P(x)$
as ...
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Mixing Continued fraction and nested radicals does $D=-\pi+\ln(16)+(7\ln\pi)/2-7/3\arctan(\pi)$?
Let :
$$B=\cdot\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\cdots}}}}}}}}$$
Let :
$$D=\frac{1}{B+\frac{1}{B+\frac{1}{B+\frac{1}{B+\cdots}}}}$$
Then I conjecture with WA that ...
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How to solve $\sin\left(1+\frac{1+\sqrt{5}}{2}\right)<1/2$?
Problem :
Find a geometric construction or a proof by hand to show :
$$\sin\left(1+\frac{1+\sqrt{5}}{2}\right)<1/2$$
As attempt I introduce the inequality :
$$\sin\left(1+\frac{1+\sqrt{5}}{2}\right)...
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Any cool ways to solve $\int_{0}^{\frac{\pi}{2}}\log(1+4\sin^2(x))\,dx$
As per the title, I have solved the following integral
$$\int_{0}^{\frac{\pi}{2}}\log(1+4\sin^2(x))\,dx$$
I would love to see any insights, and solution processes anyone may have in solving it as well....
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The golden ratio in a parabola
It is nice that the golden ratio appears automatically when we are not looking for it. This is what happened to me when I was using GeoGebra and trying to solve a different problem that occurred to me:...
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What are the known methods of drawing a golden rectangle with a ruler and compass?
Googling and Wikipedia, gives only the following construction
A golden rectangle can be constructed with only a straightedge and compass in four simple steps:
Draw a square.
Draw a line from the ...
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Proof that the sequence $x_n=f_{n+1}/f_n$ convergences to the golden ration $\Phi$
I'm supposed to proof that the sequence $x_n:=f_{n+1}/f_n,n\in\mathbb{N}$ converges to the golden ratio $\Phi$ where $f_n:=f_{n-1}+f_{n-2}$ is the Fibonacci sequence with $f_1=1$ and $f_2=1$. At first ...
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Nested radicals $\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$ then $\lim_{x\to\infty}(g(x+1)-g(x))=^?1$
Well let the problem first :
Conjecture :
Let $x>M>0$, $b=\sqrt{x}$,$0<C<1$ then define :
$$\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$$
Then it seems we have :...
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How to construct (with straightedge and compass) this diagram with four circles and two chords?
Inspired by this Sangaku-style question about a constellation of circles, I've come up with the following question.
How can we construct, with straightedge and compass, the following diagram?
...
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Self-made Sangaku-style geometry problem involving chords and inscribed circles
In the diagram, circles (or disks, if you like) of the same color have the same radius. (For an explicit description of the diagram, see below.)
Let $g=$ radius of the green circles, $r=$ radius of ...
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Solving for $x$ in $x^5+x^4+x^3+x^2+x+1=0$
So I was scrolling through the homepage of Youtube to see if there were any math equations that I thought that I might be able to solve when I came across this video by Blackpenredpen which was a ...
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Limit of a series of a side of the Golden Ratio
I try to find a limit of a series of a side from the rectangle of a golden ratio (Golden Cut, Goldener Schnitt). Just for fun to learn more about series. After hours of thinking I could need some help ...
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Golden number, diagonals of polygons and continued fractions
It is well know that, for a pentagon with unit side, the diagonal $\delta$ is such that
$$
\delta : 1=1:(\delta-1)
$$
so that its length is the positive solution of the equation $x^2-x-1=0$.
i.e. the ...
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$\lim_{n\to \infty}\frac{F_{n+1}^{k}}{F_{n}^{k}}=\lim_{n\to \infty}\frac{F_{n+k}}{F_n}=\phi^{k},$ true? for odd $k$'s?
I was working with continued fractions and the Fibonacci sequence.
And concluded this.
Firstly, we know $$\lim_{n \to \infty}\frac{F_{n+1}}{F_{n}}= \phi$$ and easily you could prove it is using its ...
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Different way of proving: $\sum_{n=1}^{\infty}\frac{\left(16(-1)^n+5\right)\left(\phi H_n+\frac{1}{n^3}\right)+\frac{11}{n^3}}{n^2}=\zeta(5)$
$$\sum_{n=1}^{\infty}\frac{\left(16(-1)^n+5\right)\left(\phi H_n+\frac{1}{n^3}\right)+\frac{11}{n^3}}{n^2}=\zeta(5)\tag1$$
$\phi=\frac{1+\sqrt{5}}{2}$
$H_n=\sum_{k=1}^{n}\frac{1}{k}$
we expanded $(1)$
...
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If $a^2-a-1=0$ where $a\gt0$, then what does $a^6$ equal? (Olympiad question)
$\color{white}{\require{cancel}{3}}$
So I was looking on Youtube for math equations that I thought that I could probably solve when I came across this video by the channel Maths and many more. The ...
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How to proof of $a_{n+34}-a_{n}\ne 55$
Let two sequences:
\begin{align}
(a_n)_{n\ge 0}\;&|\;a_n=\lfloor n\phi\rfloor\\
(b_n)_{n\ge 1}\;&|\;b_n=\sum_{k=1}^n \left\lfloor 34\phi^{a_k-a_{k-1}}\right\rfloor
\end{align}
where $\phi=\...
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Why does there seem to be a relation between the fifth roots of $32$ and $\varphi$ (the golden ratio)?
I have the following problem on one of my assignments:
find all five fifth roots of 32.
What I find interesting is how $\varphi$ is related to the real component of most of the solutions. My ...
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Show an integral inequality $\int_{0}^{\phi}f\left(x^{\phi}\right)+f\left(\phi^{\phi}-x\right)dx<\phi$
Denote by $\phi$ the golden ratio then show the inequality :
$$\int_{0}^{\phi}f\left(x^{\phi}\right)+f\left(\phi^{\phi}-x\right)dx<\phi$$
Where :
$$f\left(x\right)=\left(\frac{1}{x+1}\right)^{\frac{...
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Is there any method other than Feynman’s trick which can deal further with powers higher than 2?
Background
When I met the integral
$$\int_0^1 \frac{\left(x^\phi-1\right)^2}{\ln ^2 x} d x\\$$
where $\phi$ is the golden ratio: $\phi^2= \phi+1, $
I was surprised by its simple and decent value ...
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For Which $a$ In $[0,1]^n$ Is $\text{frac}(ta)$ Maximally Aperiodic
I know that there is no 'nice' solution to this question, but I'm just curious if there exists a solution at all...
Let $c$ be the $n$-cube $[0,1]^n$.
Given a vector $a\in c$, we can look at the value ...
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What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? [duplicate]
What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? $F_n$ denotes the $n^{th}$ Fibonacci number.
Wolframalpha gave me this answer: $$\sum_{n=1}^{\infty}\frac{1}{F_n}\ =\frac{1}{4}\...
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A connection between Fibonacci numbers and the golden ratio
I'm very interested in studying number theory and I post an other conjecture between the golden ratio and Fibonacci numbers.
N.B: it is well known that is a link between the golden ratio and Fibonacci ...
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Why is $\phi^x=\underset{n\to\infty}{\lim} \frac{F_n}{F_{n-x}}$? [closed]
The function $F_n$ denotes the nth Fibonacci number and $\phi$ is the golden ratio $\frac{1+\sqrt{5}}{2}$. I found this while trying to create a fun math puzzle. Is there a name for this? Also, how do ...
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Relation between $\phi$ and $\pi$ [closed]
I saw this $$\frac 65 \phi^2 \sim \pi$$ with $99.9985\%$ of accuracy. There are many more estimations like this between $\phi $ and $\pi$, I think?!. Now the question is about the origination of this(...
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Pole of the golden spiral
Starting from a golden triangle (an isosceles triangle with the vertex angle half the basis angle) a well known construction gives a spiral (that approximate a logarithmic spiral), as we can see in ...
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Very strange limit involving factorial and golden ratio
It's a complex limit I don't understand involving the factorial and golden ratio this is :
$$\lim_{x\to 0^-}\frac{\left(1-x!^{x^{x!}}-x^{x!^{x}}\right)}{x^{1.1}\ln\left(x\right)}=-\frac{i\left(-1+\...
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Proving $\left\lfloor(\frac{1+\sqrt{5}}{2})^{4n+2}\right\rfloor-1$ is a perfect square for $n=0,1,2,\ldots$
Let $$S_n = \left \lfloor\left(\frac{1+\sqrt{5}}{2}\right)^{4n+2}\right\rfloor-1$$ ($n=0, 1, 2, \ldots$).
Prove that $S_n$ is a perfect square.
In Art of Problem Solving website, there is a hint
$$
\...
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Prove that $\lfloor {(\frac{\sqrt{5}+1}{2})}^{4n-2}\rfloor-1$ is a square number where $n$ is a natural number.
I found this problem in a junior high school math competition.
Prove that $\lfloor {(\frac{\sqrt{5}+1}{2})}^{4n-2}\rfloor-1$ is a
square number where $n$ is a natural number.
Here's what I think
...
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Numbers Whose Multiples Don't Get Near To Each Other Modulo 1
Let $d(x,y):=\min(x-y-\lfloor x-y\rfloor,y-x-\lfloor y-x\rfloor)$ (think about the geodesic distance of points on the circle), for each real number $a$ there is the monotone sequence $(\min_{0\le i<...
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Converting between arbitrary real-valued base systems
I watched a video from Combo Class on YouTube about non-integer base systems which is something I've expressed interest in before but this did get me thinking about them again. In the video we are ...
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Negative golden ratio $(–φ)$ as a number system base?
As negative numbers can be used as bases for numeral systems (e.g. negadecimal), and non-integers such as the golden ratio $φ$ can also be used as bases, I have tried to find information on whether ...
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If $\phi=\frac12(1+\sqrt5)$ and $n=\frac11+\frac1{1+\phi}+\frac1{1+\phi+\phi^2}+\cdots$, then evaluate $\lfloor2n\rfloor+\lceil2n\rceil$
The question is
Given that $\phi=\frac{1+\sqrt{5}}{2}$. Let $$n=\frac{1}{1}+\frac{1}{1+\phi}+\frac{1}{1+\phi+\phi^2}+\frac{1}{1+\phi+\phi^2+\phi^3}+\dots$$
The value of $\lfloor2n\rfloor+\lceil2n\...
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Is this general nested radical for $\pi$ true?
We have,
I. Liu Hui (c. 300 AD)
$$\pi \approx 3\cdot2^{\color{red}8}\times \underbrace{\sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+\sqrt{\color{blue}1}}}}}}}...
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Solve for golden ratio value using recursion?
The golden ratio is defined as:
$$\frac a b = \frac {a+b} a.$$
It comes down to:
$$\varphi = 1 +\frac 1 \varphi.$$
Is there a way to solve for $\varphi$ computationally/recursively? Is there any other ...
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How do I use induction to show that the $n$th term in the Fibonacci sequence is always less than or equal to $\left(\frac{1+\sqrt 5}{2}\right)^{n-1}$?
I’m pulling this question from Invitation to Discrete Mathematics by Jiri Matousek and Jaroslav Nesetril.
$\def\pgr{\left(\frac{1+\sqrt{5}}{2}\right)}$
They ask in an exercise to use induction to show ...
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Binet's Fibonacci Number Formula on the (real) Cartesian plane [closed]
Binet's Fibonacci Number Formula $\frac{φ^{n}-(-φ)^{-n}}{\sqrt 5}$ produces a sine-like curve for positive numbers and a spiral for negative numbers when plotted on a complex plane (see figs. 1., and ...
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Where do I go from here to solve $F_{N}<\phi^N$, with $\phi = \frac{1+\sqrt{5}}{2}$
The problem specified is to prove by induction the formula $$F_{N}<\phi^N,$$ with $$\phi = \frac{1+\sqrt{5}}{2}$$
So far I have proven the base case for N=1. My induction step is $$F_{k+1}<\phi^{...
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Find all functions for which their inverse equals the derivative
I was wondering how you could find all functions $f$ for which it holds true that $f^{-1} = f'$.
I found the solution $\varphi^{1-\varphi}x^{\varphi}$ for which this holds true. Are there any other ...
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If $\sum\limits_{n=1}^{\infty}e_n x^n = 0$ always implies $\sum\limits_{n=1}^{\infty}e_n a_n = 0$, then $(a_n) = (x^n)$?
Let $x \in \left[\frac{2}{\sqrt{5}+1}, 1\right)$ and assume the real sequence $(a_n)$ satisfies
$a_1 = x,$
For every sequence $(e_n) \subset \left\{-1, 0, 1 \right\}$ with $\sum\limits_{n=1}^{\infty}...
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How does $2\pi(1-\frac{1}{\phi})$ become $\pi(3 - \sqrt 5)$, where $\phi$ is the Golden Ratio?
Help me please to understand how does the Golden Angle on this mathworld page is derived: https://mathworld.wolfram.com/GoldenAngle.html
I can't understand how does it transformed from $2\pi(1-\dfrac{...
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