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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 \(n\ge 2\), \[{ F }_{ n+1 }={ F }_{ n }+{ F }_{ n-1 }\\ \] 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
11 months, 4 weeks ago

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Wow,very helpful!Thank you! Rohit Udaiwal · 11 months, 4 weeks ago

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@Rohit Udaiwal You are welcome! Did you like it? Kuldeep Guha Mazumder · 11 months, 4 weeks ago

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@Kuldeep Guha Mazumder Yes! Rohit Udaiwal · 11 months, 4 weeks ago

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@Rohit Udaiwal My pleasure..:-) Kuldeep Guha Mazumder · 11 months, 4 weeks ago

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I like Fibonacci very much.It is really The beauty of Mathematics. Gm Kibria · 9 months, 3 weeks ago

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Very nice knowledge.. Loved it...The Magic of Maths!!! Aryan Vyas · 10 months, 1 week ago

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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 (: Hummus A · 10 months, 1 week ago

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@Hummus A I have seen your proof. Your idea is essentially the same. Only some of your steps are erroneous. Kuldeep Guha Mazumder · 10 months, 1 week ago

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@Kuldeep Guha Mazumder why?can you elaborate please? Hummus A · 10 months, 1 week ago

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@Hummus A Nothing as such. Only that you have put a plus sign in front of 1/phi. Kuldeep Guha Mazumder · 10 months, 1 week ago

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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. Kushagra Sahni · 11 months, 2 weeks ago

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@Kushagra Sahni Ok then I will write a note on it.. Kuldeep Guha Mazumder · 11 months, 2 weeks ago

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@Kuldeep Guha Mazumder Didn't you find it extremely interesting? This is the beauty of Mathematics. Kushagra Sahni · 11 months, 2 weeks ago

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Nice Suryadeep Bandyopadhyay · 11 months, 3 weeks ago

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@Suryadeep Bandyopadhyay Thanks..don't you think whatever is written above is a reconciliation of two apparently different mathematical ideas?.. Kuldeep Guha Mazumder · 11 months, 3 weeks ago

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Nice work ! I read this in the book Da Vinci Code by Dan Brown. Raven Herd · 11 months, 3 weeks ago

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@Raven Herd That is one book that I want to read but haven't read yet..thank you for your compliments..:-) Kuldeep Guha Mazumder · 11 months, 3 weeks ago

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@Kuldeep Guha Mazumder Have you read any other book by Dan Brown ? If not then try them ,they are awesome . Raven Herd · 11 months, 3 weeks ago

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@Raven Herd I have just bought The Da Vinci Code today..:-) Kuldeep Guha Mazumder · 11 months, 3 weeks ago

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