Golden Ratio and Fibonacci Numbers

Golden Ratio is considered to be one of the greatest beauties in mathematics. Two numbers $$a$$ and $$b$$ are said to be in Golden Ratio if $a>b>0,\quad and\quad \frac { a }{ b } =\frac { a+b }{ a }$ If we consider this ratio to be equal to some $$\varphi$$ then we have $\varphi =\frac { a }{ b } =\frac { a+b }{ a } =1+\frac { b }{ a } =1+\frac { 1 }{ \varphi }$ Solving in quadratic we get two values of $$\varphi$$, viz. $$\frac { 1+\sqrt { 5 } }{ 2 }$$ and $$\frac { 1-\sqrt { 5 } }{ 2 }$$ one of which (the second one) turns out to be negative (extraneous) which we eliminate. So the first one is taken to be the golden ratio (which is obviously a constant value). It is considered that objects with their features in golden ratio are aesthetically more pleasant. A woman's face is in general more beautiful than a man's face since different features of a woman's face are nearly in the golden ratio.

Now let us come to Fibonacci sequence. The Fibonacci sequence ${ \left( { F }_{ n } \right) }_{ n\ge 1 }$ is a natural sequence of the following form:${ F }_{ 1 }=1,\quad { F }_{ 2 }=1,\quad { F }_{ n-1 }+{ F }_{ n }={ F }_{ n+1 }$ The sequence written in form of a list, is $1,1,2,3,5,8,13,21,34,..$.

The two concepts: The Golden Ratio and The Fibonacci Sequence, which seem to have completely different origins, have an interesting relationship, which was first observed by Kepler. He observed that the golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence or any Fibonacci-like sequence (by Fibonacci-like sequence, I mean sequences with the recursion relation same as that of the Fibonacci Sequence, but the seed values different). In terms of limit:$\underset { n\rightarrow \infty }{ lim } \left( \frac { { F }_{ n+1 } }{ { F }_{ n } } \right) =\varphi$ We shall now prove this fact. Let ${ R }_{ n }=\frac { { F }_{ n+1 } }{ { F }_{ n } } ,\forall n\in N$Then we have $\forall n\in N$ and $n\ge 2$, ${ F }_{ n+1 }={ F }_{ n }+{ F }_{ n-1 }\$/extract_itex] and ${ R }_{ n }=1+\frac { 1 }{ { R }_{ n-1 } } >1$We shall show that this ratio sequence goes to the Golden Ratio $\varphi$ given by: $\varphi =1+\frac { 1 }{ \varphi }$We see that: $\left| { R }_{ n }-\varphi \right| =\left| \left( 1+\frac { 1 }{ { R }_{ n-1 } } \right) -\left( 1+\frac { 1 }{ \varphi } \right) \right| \\ =\left| \frac { 1 }{ { R }_{ n-1 } } -\frac { 1 }{ \varphi } \right| \\ =\left| \frac { \varphi -{ R }_{ n-1 } }{ \varphi { R }_{ n-1 } } \right| \\ \le \left( \frac { 1 }{ \varphi } \right) \left| \varphi -{ R }_{ n-1 } \right|\\ \le { \left( \frac { 1 }{ \varphi } \right) }^{ n-2 }\left| { R }_{ 2 }-\varphi \right|$ Which clearly shows that$\left( { R }_{ n } \right) \longrightarrow \varphi$ (since $\left| { R }_{ 2 }-\varphi \right|$ is a finite positive real whose value depends on the seed values) Note by Kuldeep Guha Mazumder 4 years, 8 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. 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 1paragraph 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} $\frac{2}{3}$
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- 4 years, 8 months ago

You are welcome! Did you like it?

- 4 years, 8 months ago

Yes!

- 4 years, 8 months ago

My pleasure..:-)

- 4 years, 8 months ago

Nice work ! I read this in the book Da Vinci Code by Dan Brown.

- 4 years, 7 months ago

That is one book that I want to read but haven't read yet..thank you for your compliments..:-)

- 4 years, 7 months ago

Have you read any other book by Dan Brown ? If not then try them ,they are awesome .

- 4 years, 7 months ago

I have just bought The Da Vinci Code today..:-)

- 4 years, 7 months ago

Nice

- 4 years, 7 months ago

Thanks..don't you think whatever is written above is a reconciliation of two apparently different mathematical ideas?..

- 4 years, 7 months ago

There is one more interesting thing I found yesterday. The Ratio of the diagonal and the side of a regular Pentagon is exactly equal to the golden ratio.

- 4 years, 7 months ago

Ok then I will write a note on it..

- 4 years, 7 months ago

Didn't you find it extremely interesting? This is the beauty of Mathematics.

- 4 years, 7 months ago

https://brilliant.org/problems/wow-12/?group=w3HWB8GobVLl&ref_id=1095702

i posted a problem about the same thing

my solution was almost the same as your proof of it (:

- 4 years, 6 months ago

I have seen your proof. Your idea is essentially the same. Only some of your steps are erroneous.

- 4 years, 6 months ago

- 4 years, 6 months ago

Nothing as such. Only that you have put a plus sign in front of 1/phi.

- 4 years, 6 months ago

Very nice knowledge.. Loved it...The Magic of Maths!!!

- 4 years, 6 months ago

I like Fibonacci very much.It is really The beauty of Mathematics.

- 4 years, 5 months ago

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