# Questions tagged [golden-ratio]

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

458 questions
Filter by
Sorted by
Tagged with
51 views

### 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’...
34 views

### 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 ...
46 views

### 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 ...
1 vote
74 views

### 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 ...
76 views

1 vote
77 views

### 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 ...
69 views

### 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 ...
38 views

### 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 ...
1 vote
55 views

### 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,...
114 views

### 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 ...
46 views

Denoting $\varphi=\frac{1+\sqrt{5}}{2}=\mathrm{Golden\ Ratio}$. How would you prove that : $$\int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{... 1 vote 0 answers 62 views ### Approximation to the sum of reciprocals of prime numbers squared involving \phi, the golden ratio I was interested in the prime zeta function and its values, so I calculated in Excel the sum of reciprocals of prime numbers squared up to 1 MM, and tried to relate it to some known irrational ... 1 vote 1 answer 67 views ### Why is the intersections between f(x)=x^{2}+x^{-2}-3 and the x-axis resembles \varphi? The story behind this is quite silly, I was messing around with functions in geogebra, inputting several functions to see their properties, until I was amazed by something i don't know about. ... 3 votes 3 answers 81 views ### Prove that \left \lfloor \frac{1+\lfloor na+1/a\rfloor}{a} \right \rfloor=n If a \geq \frac{1+\sqrt{5}}{2} and n \in \Bbb W, prove that$$\left \lfloor \frac{1+\left\lfloor \frac{1+na^2}{a}\right\rfloor}{a} \right \rfloor=n.$$I could prove only when a is an integer, ... 0 votes 1 answer 222 views ### golden section search algorithm: why are there two 'update' equations? I understand that the golden section search algorithm (for finding minimum points) is loosely based on the bisection method (for finding roots). In both methods, we first assign an upper and a lower ... 1 vote 2 answers 326 views ### \pi & \phi (Golden ratio), Pentagon inscribed in unit circle Everyone is aware that square inscribed in unit circle and infinite product giving rise to \pi. One of the simplest way to represent \pi with the help of nested radical as follows$$\pi = \lim_{n\...
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 ...