The **Fibonacci numbers** are well-known by the general public, due to their applications in many branches of science. If f<sub>n</sub> denotes the n-th Fibonacci number, these numbers are defined by the recurrence relation:  f<sub>n+2</sub>=f<sub>n+1</sub>+f<sub>n</sub>, with the initial data or seeds f<sub>0</sub>=0 and  f<sub>1</sub>=1. Hence the Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,....

We would like to have a closed formula for f<sub>n</sub>, i.e. a formula which does not involve  recurrence relations. The answer is given by the **Binet's formula**. It is named after the French mathematician **Jacques Philippe Marie Binet** (1786-1856) who published it in 1843, even though this formula was already known by Abraham de Moivre (1667-1754), Leonhard Euler (1707-1783) and Daniel Bernoulli (1700-1782).

The Binet's formula is 
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https://steemitimages.com/DQmUMJXFBcHCGAf1uMvw67MxgLNtPdaRxLTYXVdtXzAmkN9/bin1.png
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However there is a nicer way to express it, in terms of the the **golden mean**. Let &Phi; be the golden mean, also known as the golden ratio:
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https://steemitimages.com/DQmQ14eAXmzK25XnkwaStsgGAV94NV8dmT64kcUHwwmXb2x/bin2.png
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We recall that it is a root of the polynomial x<sup>2</sup>-x-1, this is the polynomial associated to the Fibonacci recurrence relation. Let &phi; be the other root of this polynomial, i.e.
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https://steemitimages.com/DQmYiqqEm2C4JUxAseeBRhNZB1MeZ6GL51SjKfong4WkcQC/bin3.png
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The formula (1) in terms of &Phi; and &phi; is
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https://steemitimages.com/DQmSKBZfExn61qGq48jJ7pJnVwiNuFrp6Fz4yzsNXfwYazg/bin4.png
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In the following lines we will give a proof of this formula. Let Z<sub>n</sub>=a&Phi;<sup>n</sup>+b&phi;<sup>n</sup>, where a and b are constants.

We will show that Z<sub>n</sub> satisfies the same recurrence relation than the Fibonacci numbers, i.e.  Z<sub>n</sub>=Z<sub>n-1</sub>+Z<sub>n-2</sub>, and find the values of a and b for which Z<sub>0</sub>=0 and  Z<sub>1</sub>=1. Then the identity (2) follows.

Since &Phi; and &phi; satisfy the equation x<sup>2</sup>=x+1, for n a positive integer we have:
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https://steemitimages.com/DQmbubRyb9s5sfgsApQhWvNj3vM8zKVUDiUYMR5vk3Az2te/bin5.png
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Hence
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https://steemitimages.com/DQmeJsUoTyk71C8Qr19GvBm2QXcVQqYipYPv5f68wBkGxUp/bin6.png
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If we set Z<sub>0</sub>=0, then a+b=0. Similarly if  Z<sub>1</sub>=1 then a&Phi;+b&phi;=1. So we have the system of equations:
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https://steemitimages.com/DQmWBWZyz9JJgCyYJGJxFBSibBYweNmVPrJpi2J2GWyiNzB/bin7.png
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Its solution is
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https://steemitimages.com/DQmT8vSAQBNfVyCCfDLu8FGF7gsUTMMkjoasR8n9i5n5Qdq/bin8.png
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Hence  Z<sub>n</sub>=f<sub>n</sub> and the identity (2) holds.

References:
https://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_expression
http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html
https://en.wikipedia.org/wiki/Jacques_Philippe_Marie_Binet
T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001.

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<h5>All the formulae of this post were typed by myself in LaTeX.</h5>