Questions tagged [golden-ratio]
Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$
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Proof that $\sum_{n=1}^\infty\frac{F_n}{3^n n} = \frac{\ln(\phi+1)}{\sqrt{5}}$
I conjectured by computation the following, but I’m not sure where to start to prove it.
$$\sum_{n=1}^\infty\frac{F_n}{3^n n} = \frac{\ln(\phi+1)}{\sqrt{5}}$$
where $F_n$ are the Fibonacci numbers. I’...
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What's the intuition of the relation between fibonacci-like sequences and the proportion used to obtain the golden ratio?
Everyone knows that we can obtain the golden ratio from the following proportion:
$$\frac{a}{b} = \frac{a+b}{a}$$
We also know that we get ${\phi}^N$ when we try to find a function that satisfies the ...
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Rational approximations of the golden ratio: how to prove this limit exists?
Given a positive real number $\alpha$ and a positive rational number $p/q$ in reduced form let's define the quality of $p/q$ as an approximation to $\alpha$ as$$-\log_q|\alpha - p/q|$$
I'm looking at ...
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proof fibonacci sequence is small o(2^n) without using closed formula
I need to prove that for the given fibonacci sequence,
with initial values:
f(1)=1
f(2)=2
f(n)=f(n-1)+f(n-2)
f(n) is belong to small o(2^n).
I need to prove it without using the closed formula of ...
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Why does the golden ratio emerge in this primorial-related sequence?
Let $$f(i):=\left\lfloor\frac{p_i\#}{\varphi(p_i\#)}\right\rfloor,$$
where $p_i$ is the $i$th prime, $\#$ is the primorial operator, and $\varphi$ is totient.
Example
$$f(3)=\left\lfloor\frac{5\#}{\...
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Compute $p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$
I need to compute $$p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$$ where $p=\frac{1}{2}$, $r_1= \frac{1+\sqrt5}{4}= \frac{1}{2}\varphi$ and $r_2= \frac{1-\sqrt5}{4}$.
Using properties of ...
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Non-periodic continued fraction with explicitly known convergents?
Is an irrational number with non-periodic continued fraction expansion known, for which one can give explicit formulas for the convergents $p_n/q_n$ or at least for the denominators $q_n$ (similar to ...
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Relation between logarithmic spirals and the golden ratio
The polar equation of a logarithmic spiral curve is given by
$$ r= ae^{b\theta} $$
where each point on the curve is described in polar coordinates: $r$ is the distance from the origin and $\theta$ is ...
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What does the golden ratio have to do with complex hyperbola and real circle
If you have $xy = i $ and $x^2 + y^2 = 1$ then you get the solutions that have the golden ratio in them.
These are the solutions
Wolfram calculation:
https://www.wolframalpha.com/input/?i=xy%3Di%2C+x%...
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Is my proof for the Irrationality of the Golden Ratio correct?
I am trying to make a proof about the irrationality of the golden ratio by contradiction. I substituted the fraction $\frac{p}{q}$ for $\varphi$, where p and q are integers that don't have a common ...
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L and M are midpoints of equilateral triangle ABC, and LM meets the circumcircle of the triangle at Y. Prove LM/MY is the golden ratio.
So far I've tried to split the triangle into smaller 30-60-90 triangles, and letting LM = x make some proportions and see if they can be related in anyway. I'm not sure what to do about MY though.
Any ...
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Dodecahedron and golden ratio algebra
We can see that the volume of a dodecahedron of size $2\varphi$, where $\varphi=\frac{\sqrt{5}-1}{2}$, can be found in two ways. The first one uses the pentagonal pyramids with the faces as basis and ...
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Equations for half-integer points on generalized complex Fibonacci sequence (metallic mean sequence)
I have been experimenting with generalizing the Fibonacci sequence, and Fibonacci-like "metallic mean" sequences such as the Pell sequence, to non-integer and complex values.
The standard, ...
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Golden ratio in complex number squares
In the Argand diagram shown below the complex numbers
$– 1 + i, 1 + i, 1 – i, – 1 – i$ represent the vertices of a square ABCD.
The equation of its diagonal BD is $y = x$. The complex number $k + ki$
...
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Exercise 0.21 in Miles Reid's Commutative Algebra
I have the following exercise in Miles Reid's Undergraduate Commutative Algebra:
Exercise 0.21: Consider the ring $B′=\Bbb Z[\tau]$, where $\tau^2=\tau+1$. Show that an element $a+b\tau$ is a unit of ...
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For which recursive functions (or infinite expressions) can substitution be used for a solution?
I have a question about recursive functions and their convergence as sequences.
We know that the golden ratio can be represented by:
$$
\phi= 1+\frac{1}{1+\frac{1}{1+\frac{1}{...}}}
$$
Rewriting the ...
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why is the golden ratio algebraic and the golden angle transcendental [closed]
I'm referencing the golden angle from here https://en.wikipedia.org/wiki/Golden_angle, where it's defined as $ 2\pi(1-\frac{1}{\varphi})$.
All constructible numbers are algebraic numbers.
A ...
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How to solve $\left(\sqrt{\frac{x-1}{x}}\right)^{x^2}=\left(\frac{1}{x}\right)^{x+1}$?
I need help to solve this equation, please.
$$\left(\sqrt{\frac{x-1}{x}}\right)^{x^2}=\left(\frac{1}{x}\right)^{x+1}$$
I know that the solution is $x=\varphi$ (the golden ratio). I got this result by ...
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Is there a function for $[1;x,x,...]$ (as an infinite continued fraction)?
So I was experimenting with continued fractions and came up with an idea for a function $$f(x) =1+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\dots}}}}}},$$ as an infinite ...
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How are Ostrowski Numeration Systems built from a periodic continued fraction?
Ostrowski number systems represent integers and real numbers using the denominators of the convergents of continued fractions. One better known special case of this is Zeckendorf's Theorem and related ...
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A Golden Angle Conjecture
Update: this conjecture is broken, for n=7. See final note.
This conjecture is related to the process of phyllotaxis in plants, and understanding why nature would choose to iterate the Golden Angle ...
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Julia Set Fixed Points And The Golden Ratio
When calculating fixed points for the Basilica Julia set ($z_{n+1}=z^2+c$, $c=-1$), the fixed points are given by
$$
z^*_0(-1)=\frac{1 \mp \sqrt{1-4(-1)}}{2}=\frac{1 \mp \sqrt{5}}{2}
$$
This is ...
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Fibonacci golden ratio question without assuming convergence a priori
Prove $\frac{F(n+1)}{F(n)}$ converges to $\phi$ without assuming a priori that it converges
If I know it converges, then I know it converges to $\phi$
Since, if $\lim_{n\to\infty} \frac{F(n+1)}{F(n)} \...
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Approximation of the golden ratio using exponential
Claim :
$$\left(1+e^{-1}\right)^{3}\cdot\left(1-e^{-1}\right)-\frac{\left(\sqrt{5}+1\right)}{2}<0$$
I found it accidentaly using Desmos .It's quite good as approximation of the golden ratio.
Some ...
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Rectangles with side lengths proportional to transcendental numbers BESIDES $\varphi$
A rectangle with side lengths proportional to $\varphi$ is a "golden rectangle". If we perform a procedure of subtracting the shorter side length from the longer side length, a golden ...
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Derive an exact value for the Golden ratio
Have I accurately solved with the given information?
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Why does the golden ratio show up as the solution to these two simultaneous equations?
I was given the following simultaneous equations to solve on a homework sheet:
$$
x^2 + y^2 = 3\\x-y=1
$$
And when I did so I got the answers of:
$$
(\varphi,-1/\varphi)\\
(1/\varphi,-\varphi)
$$
I ...
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Is there a notion of Lucas/Fibonacci sequences beyond adding two numbers at a time, if so where could I read more about that?
So I recently learned about lucas/fibbonaci sequences EG; 1,1,2,3,5,7 or 3,4,7,11,18... A fact I learned about these sequence is that the limit of the quotient of the terms approaches the golden ratio ...
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Estimating golden ration by simulation
I've been looking at Monte Carlo simulation recently, and have been using it to approximate constants such as π (circle inside a rectangle, proportionate area).
However, I'm unable to think of a ...
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Fibonacci generating function - quick way to see formula
I know that
$$
\frac{1}{1-z-z^2} = \sum_{k = 1}^\infty a_nz^n
$$
in $B_{\varepsilon}(0) \subseteq \mathbb{C}$ where $(a_n)_{n \in \mathbb{N}}$ denotes the Fibonacci series and $\varepsilon > 0$ is ...
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Improper integral inequality including the golden ratio and the Sophomore's dream
It's an inequality I found nice let me propose it :
Let $0\leq x$ then we have :
$$\int_{0}^{\infty}\sin\left(x^{-x}\right)dx<\phi=\frac{\left(1+\sqrt{5}\right)}{2}\quad (I)$$
My attempt :
First ...
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Bounding the sum of a floor function involving the golden ratio
Consider the sum
\begin{equation*}
S_n = \displaystyle \sum_{k=1}^n \left( k\varphi - \left\lfloor k\varphi\right\rfloor -\frac12 \right)
\end{equation*}
where $\quad\varphi=\dfrac{1+\sqrt{5}}{2}\quad$...
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Exercise 0.20 in Miles Reid's Commutative Algebra
I have the following exercise in Miles Reid's Undergraduate Commutative Algebra:
Exercise 0.20: Consider the ring $B'=\Bbb Z[\tau]$, where $\tau=\frac{1+\sqrt 5}{2}$ is the so called "golden ...
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$a^2\phi=2$ , what is the value of $2a(\phi+1)$?
I reduced a problem to this:
We have $a^2\phi=2$ where $a>0$ what is the value of $2a(\phi+1)$ ?
$1)2\sqrt{2\sqrt5+4}\qquad\qquad2)2\sqrt{\sqrt5+4}\qquad\qquad3)2\sqrt{2\sqrt5+1}\qquad\qquad4)2\...
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Prove that $\int _{-\infty }^{+\infty }{\frac {\mathrm {d} z}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}=1$
Recently while dealing with few interesting integrals, I was quite fascinated by this one:
$$\int _{-\infty }^{+\infty }{\dfrac {1}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\...
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Prove without induction that the sequence $a_{n}=1+\frac{1}{a_{n-1}}$ $=$ $\frac{F_{n+3}}{F_{n+2}}$ with $a_1=2$ where $F_n$ is the Fibonacci sequence
While trying to prove that the sequence $a_{n+1}=1+\frac{1}{a_n}$ with $a_1=2$ converges to Phi(the second sequence obviously converges to Phi), I recognized the pattern when I calculated the first ...
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Proof if the digits of the golden ratio are in sequence in the digits of Pi or not [closed]
Since the digits of $\pi$ are uniformly random and infinite, any finite sequence of digits can be found in sequence somewhere among the digits of $\pi$.
But does this also holds when it comes to ...
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How deep into a Fibonacci matrix would be needed to calculate pi to 100 digits?
(0,1),1,2,3
(1,1),2,3,5
(2,1),3,4,7
(3,1),4,5,9
Then use Wallis product, as a wave descending the layers, to calculate pi in various ways such as pi/2 =
2/1 x 2/3 x 4/3 x 4/5 x...
Also able to ...
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Is it true that $f(x)^{f(x)^{\dots}}$ with $f(x) = -x^2 -x+1$ converges to the Golden Ratio? Why?
My son accidentally discovered and, given $f(x)=-x^{2}-1x+1$, $f\left(x\right)^{f\left(x\right)^{f\left(x\right)^{f\left(x\right)}}}$ intersects 1.0 (i.e., $y=1.0$) at the negative of the golden ratio,...
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iterated function $a(n) = \lfloor n\phi + 0.5\rfloor$
Let $a(n) = \lfloor n\phi + 0.5\rfloor$ for all $n \geqslant 1$, where $\phi$ is the golden ratio. Now let
$$a(n)^k = a(a(\ldots(a(n))))$$
where we have iterated $a(n)$, $k$ times in the RHS. I am ...
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Fibonacci number of local minima in sum of sines
I have this function
$$\begin{align}f_N:[-1,1]&\longrightarrow [-1,1] \\
\hspace{-10mm}x &\longrightarrow \frac{1}{N}\sum_{i=0}^N cos(\phi^i \pi x) \end{align}$$
where $\phi=\frac{1+\sqrt 5}{2}...
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Golden Exponent? Tetration
I read somebody say “golden exponent $ x^x=x+1 $” now I didn’t understand what he meant but it really fascinated me thinking about some type of tetration version of the golden number. A number with a ...
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Given a recursion $a_{n+ 1}= a_{n}^{2}+ 1$ with $a_{0}> 0,$ I want to consider the asymptotic behavior of this sequence.
Given a recursion $a_{n+ 1}= a_{n}^{2}+ 1$ with $a_{0}> 0,$ I want to consider the asymptotic behavior of this sequence.
Till now
$$a_{n}> a_{n- 1}^{2}> \cdots> a_{0}^{2^{n}}$$
and for $\...
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Prove that : $ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{5}\zeta\left(3\right) $
Denoting $ \varphi=\frac{1+\sqrt{5}}{2}=\mathrm{Golden\ Ratio} $. How would you prove that : $$ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{...
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Approximation to the sum of reciprocals of prime numbers squared involving $\phi$, the golden ratio
I was interested in the prime zeta function and its values, so I calculated in Excel the sum of reciprocals of prime numbers squared up to $1$ MM, and tried to relate it to some known irrational ...
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Why is the intersections between $f(x)=x^{2}+x^{-2}-3$ and the x-axis resembles $\varphi$?
The story behind this is quite silly, I was messing around with functions in geogebra, inputting several functions to see their properties, until I was amazed by something i don't know about.
...
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Prove that $\left \lfloor \frac{1+\lfloor na+1/a\rfloor}{a} \right \rfloor=n$
If $a \geq \frac{1+\sqrt{5}}{2}$ and $n \in \Bbb W$, prove that
$$\left \lfloor \frac{1+\left\lfloor \frac{1+na^2}{a}\right\rfloor}{a} \right \rfloor=n.$$
I could prove only when $a$ is an integer, ...
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golden section search algorithm: why are there two 'update' equations?
I understand that the golden section search algorithm (for finding minimum points) is loosely based on the bisection method (for finding roots). In both methods, we first assign an upper and a lower ...
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$\pi$ & $\phi$ (Golden ratio), Pentagon inscribed in unit circle
Everyone is aware that square inscribed in unit circle and infinite product giving rise to $\pi$.
One of the simplest way to represent $\pi$ with the help of nested radical as follows
$$\pi = \lim_{n\...
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Nested radicals, Golden ratio, number series
Finite and infinite expansion of nested Radical involving Golden ratio as follows
$2\cos\frac{2\pi}{5} = \frac{1}{\phi} $ where $\phi$ is Golden ratio $(2\cos72°)$
Let us expand this with increasing ...