# 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 $\sin (\phi+1)<\frac12$, ...
1 vote
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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 ...
204 views

### 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 : $$\phi=f(f(\cdots f(\phi)\cdots))$$ ...
153 views

### 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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### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...