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I have read in a page that " To find the nth term of the Fibonacci series, we can use Binet's Formula"

F(n) = round( (Phi ^ n) / √5 ) provided n ≥ 0
where Phi=1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ... .

my question is,

  • what is this Phi?
  • how they generated this Phi?
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up vote 4 down vote accepted

$\phi = \frac{1+\sqrt{5}}{2}$ is a very important constant in nature called the golden ratio, it appears in many different guises.

Binets formula is:

$F_n = \frac{(\phi^n - (-\phi)^{-n})}{\sqrt{5}} = \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n \sqrt{5}}$

There are many different ways to prove Binet's formula. One I like in particular (which generalises to other linear difference equations) is to use eigenvalues and eigenvectors of a particular matrix. This is essentially the reason behind the appearance of $\phi$.

See here:

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The $\varphi$ here is the famous Golden ratio. It appears to be one of those fundamental numbers that arises from the nature itself. The Wikipedia link above provides a good discussion about its nature and its occurences in math.

As for its particular connection to the Fibonacci numbers, Wikipedia again gives a good explanation. (From a philosophical perspective, I would actually say that Fibonacci numbers are more fundamental than $\varphi$, but this is just a personal feeling...)

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Maybe you can take a look at this : It is explain how you can calculate the $n^{th}$ term and give a method to calculate $\phi$.

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