Questions tagged [golden-ratio]
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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’... sequences-and-series generating-functions closed-form fibonacci-numbers golden-ratio- 3,195
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 ... sequences-and-series fibonacci-numbers golden-ratio- 1
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 ... sequences-and-series fibonacci-numbers golden-ratio- 51
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 ... computational-complexity fibonacci-numbers golden-ratio- 11
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 $$f(3)=\left\lfloor\frac{5\#}{\... sequences-and-series elementary-number-theory prime-numbers golden-ratio primorial- 4,993
Compute $p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$
I need to compute $$p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$$ where $p=\frac{1}{2}$, $r_1= \frac{1+\sqrt5}{4}= \frac{1}{2}\varphi$ and $r_2= \frac{1-\sqrt5}{4}$. Using properties of ... elementary-number-theory summation fibonacci-numbers golden-ratio- 41
Non-periodic continued fraction with explicitly known convergents?
Is an irrational number with non-periodic continued fraction expansion known, for which one can give explicit formulas for the convergents $p_n/q_n$ or at least for the denominators $q_n$ (similar to ... irrational-numbers continued-fractions golden-ratio- 33
Relation between logarithmic spirals and the golden ratio
The polar equation of a logarithmic spiral curve is given by $$ 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 ... logarithms golden-ratio- 1,354
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: circles hyperbolic-functions golden-ratio- 353
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 ... elementary-number-theory golden-ratioL 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 ... geometry triangles circles golden-ratio- 145
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 ... algebra-precalculus polyhedra golden-ratio- 60.9k
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, ... complex-numbers fibonacci-numbers golden-ratio- 751
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 $k + ki$ ... linear-algebra complex-numbers golden-ratio- 1,033
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 ... abstract-algebra commutative-algebra solution-verification golden-ratio- 11.5k
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