Questions tagged [golden-ratio]

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

The golden ratio is defined to be the (unique) positive number $\varphi$ for which

$$\frac{\varphi + 1}{\varphi} = \frac{\varphi}{1}$$

or alternatively, the unique positive solution of

$$x^2 - x - 1 = 0$$

It can be written exactly as

$$\varphi = \frac{1 + \sqrt{5}}{2}$$

This number has been studied since antiquity, and the quantity frequently occurs in nature and art. It is also closely related to the Fibonacci numbers.

Reference: Golden ratio.

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Approximation for $\pi$

I just stumbled upon $$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$ which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I wonder if it has been every used in past times…
John Alexiou
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Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

Let $\varphi=\frac{1+\sqrt5}2$ (the golden ratio). How can I simplify the following expression? $$7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$$
OlegK
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Continued fraction involving Fibonacci sequence

What is the limit of the continued fraction: $$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}\ $$ that involves the Fibonacci sequence terms as denominators? I've been looking for this specific continued…
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How to find $\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n$

Here I mean the limit of the following sequence: $$p_1=\int_0^1 \sqrt{x} ~dx=\frac{2}{3}$$ $$p_2=\int_0^1 \int_0^1 \sqrt{x+\sqrt{y}} ~dxdy=\frac{8}{35}(4 \sqrt{2}-1) = 1.06442\dots$$ $$p_3=\int_0^1 \int_0^1 \int_0^1 \sqrt{x+\sqrt{y+\sqrt{z}}}…
Yuriy S
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Why does every "fibonacci like" series converge to $\phi$?

It's is well known that the ratio of side-by-side fibonacci numbers converge to $\phi$. But it seems by my calculations, that if one starts with any pair of numbers one will also get a ratio that converges to $\phi$. Say for example if one starts…
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Fibonacci sequence and other metallic sequences emerged in the form of fractions

The Fibonacci sequence $P_n = P_{n-1}+P_{n-2}$ is $$1,1,2,3,5,8,13,21,34,55,89,144,233,377, 610, \cdots $$ I learnt that the fraction $1/89$ contains all the numbers in the sequence. $$\begin{align} \frac{1}{89}&=…
Larry
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On the formula, $\pi = \frac 5\varphi\cdot\frac 2{\sqrt{2+\sqrt{2+\varphi}}}\cdot\frac 2{\sqrt{2+\sqrt{2+\sqrt{2+\varphi}}}}\cdots$

I found a formula on google images when I was looking at some formulas for $\pi$ just for the fun of it, and I came across one that really startled me, and was quite reminiscent of Viète's product. Let $\varphi = \cfrac{1+\sqrt 5}2$ then $$\pi =…
Mr Pie
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Prove that the golden ratio is irrational by contradiction

I am struggling to see where the contradiction lies in my proof. In a previous example, $1/\phi = \phi-1$ where $\phi$ is the golden ratio $\frac{\sqrt{5} + 1}{2}$. Since I am proving by contradiction, I started out by assuming that $ϕ$ is…
Michelle Drolet
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Polylogarithm ladders for the tribonacci and n-nacci constants

While reading about polylogarithms, I came across the nice polylogarithm ladder, $$6\operatorname{Li}_2(x^{-1})-3\operatorname{Li}_2(x^{-2})-4\operatorname{Li}_2(x^{-3})+\operatorname{Li}_2(x^{-6}) = \frac{7\pi^2}{30}\tag{1}$$ where $x = \phi =…
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Golden Rectangle into Golden Rectangles

Can these golden rectangles be rearranged to exactly cover the underlying cyan golden rectangle? That's the entire question. All that follows is related discussion. I want to make a more elegant proof without words. It comes from the series:…
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Charming approximation of $\pi$: $2\left(\frac{1}{2}\right)^{\phi/2}+2< \pi$, where $\phi$ is the golden ratio

Prove that : $$2\left(\frac{1}{2}\right)^{\phi/2}+2< \pi$$ where $\phi:=\frac12(1+\sqrt{5})=1.618\ldots$ is the golden ratio. How I came across this approximation? Well, I was studying the following function: $$f(x)=x^{\phi(1-x)}+(1-x)^{\phi x…
Erik Satie
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The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

I am fascinated by the following formula for the golden ratio $\varphi$: $$\Large\varphi = \frac{\sqrt{5}}{1 + \left(5^{3/4}\left(\frac{\sqrt{5} - 1}{2}\right)^{5/2} - 1\right)^{1/5}} - \frac{1}{e^{2\pi\,/\sqrt{5}}}\,\mathop{\LARGE…
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How to prove $_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9\big)=\large \frac{3}{5^{5/6}}\,\phi^{-1}\,$ with golden ratio $\phi$?

(Note: This is the case $a=\frac16$ of ${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}.\,$ There is also $a=\frac13$ and $a=\frac14$.) After investigating…
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Prove that $\sum_{n=1}^\infty \left(\phi-\frac{F_{n+1}}{F_{n}}\right)=\frac{1}{\pi}$

So, I know that $$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\phi$$ where $F_n$ stands for the n'th Fibonacci number I was interested in measuring the error of the convergence of the above limit and was drawn to the conjecture that: $$\sum_{n=1}^\infty…
user311151
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Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime number appears to be. Is that correct, and if so,…
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