List questions from golden-ratio

Published on 15 Sep 2022 - 12:25

121

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How do I use induction to show that the $n$th term in the Fibonacci sequence is always less than or equal to $\left(\frac{1+\sqrt 5}{2}\right)^{n-1}$?

Published on 25 Oct 2022 - 7:51

269

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Solve for golden ratio value using recursion?

Published on 31 Oct 2022 - 4:15

206

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Is this general nested radical for $\pi$ true?

Published on 21 Nov 2022 - 3:39

129

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If $\phi=\frac12(1+\sqrt5)$ and $n=\frac11+\frac1{1+\phi}+\frac1{1+\phi+\phi^2}+\cdots$, then evaluate $\lfloor2n\rfloor+\lceil2n\rceil$

Published on 23 Nov 2022 - 11:37

219

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Negative golden ratio $(–φ)$ as a number system base?

Published on 05 Dec 2022 - 15:37

108

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Converting between arbitrary real-valued base systems

Published on 06 Dec 2022 - 6:49

54

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Numbers Whose Multiples Don't Get Near To Each Other Modulo 1

Published on 15 Jan 2023 - 14:09

107

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Prove that $\lfloor {(\frac{\sqrt{5}+1}{2})}^{4n-2}\rfloor-1$ is a square number where $n$ is a natural number.

Published on 17 Jan 2023 - 9:16

134

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Proving $\left\lfloor(\frac{1+\sqrt{5}}{2})^{4n+2}\right\rfloor-1$ is a perfect square for $n=0,1,2,\ldots$

Published on 17 Jan 2023 - 13:59

136

Views

Very strange limit involving factorial and golden ratio

Published on 20 Jan 2023 - 18:15

List Question

Binet's Fibonacci Number Formula on the (real) Cartesian plane

How do I use induction to show that the $n$th term in the Fibonacci sequence is always less than or equal to $\left(\frac{1+\sqrt 5}{2}\right)^{n-1}$?

Solve for golden ratio value using recursion?

Is this general nested radical for $\pi$ true?

If $\phi=\frac12(1+\sqrt5)$ and $n=\frac11+\frac1{1+\phi}+\frac1{1+\phi+\phi^2}+\cdots$, then evaluate $\lfloor2n\rfloor+\lceil2n\rceil$

Negative golden ratio $(–φ)$ as a number system base?

Converting between arbitrary real-valued base systems

Numbers Whose Multiples Don't Get Near To Each Other Modulo 1

Prove that $\lfloor {(\frac{\sqrt{5}+1}{2})}^{4n-2}\rfloor-1$ is a square number where $n$ is a natural number.

Proving $\left\lfloor(\frac{1+\sqrt{5}}{2})^{4n+2}\right\rfloor-1$ is a perfect square for $n=0,1,2,\ldots$

Very strange limit involving factorial and golden ratio

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