Waste less time on Facebook — follow Brilliant.
×

interesting fact about golden mean

i was solving some problem, and i found a fun facts about the golden mean. \( Golden\quad mean =\Phi=\dfrac{1+\sqrt{5}}{2}\)\[\] \[ F(n)={(((((\Phi-1)^2-2)^2-2)^2-2\dotsm}[n\quad times]\] \[ f(n)=\begin{cases} -\Phi\quad if \quad n\in [odd]\\ \Phi-1\quad if\quad n\in[even]\\ \end{cases}\] seems interesting, but why??? \[\] note that \[\Phi^2 =\Phi+1\] now, \[(\Phi-1)^2-2= \Phi^2-2\Phi+1-2=\Phi+1-2\Phi-1=\boxed{-\Phi}\] \[((\Phi-1)^2-2)^2-2=(-\Phi)^2 -2=-\Phi^2-2=-\Phi+1-2=\boxed{\Phi-1}\] \[(((-\Phi-1)^2-2)^2-2)^2 -2=(\Phi-1)^2-2=\boxed{-\Phi}\] and on and on and on, this is quite fascinating because \(\Phi\) is involved in a recurring pattern!!!

Note by Aareyan Manzoor
2 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...