Often the golden ratio is linked with the sequence of Fibonacci. Here's a confusion of tonguesbetween the 'golden ratio' as a geometrical construction and the numeber [math]\Phi[/math].[br]Then again, where does this link come from?[br]Whell, likewise [math]\Phi[/math] appears in regular pentagons and decagons because of a simple goniometrical property, there's a logic relation with the sequence of Fibonacci. This is the general formula to calclutate the n-th number of the sequence. In this formula there's not just [math]\sqrt{5}[/math], you can even rewrite the formula using [math]\Phi[/math] and [math]\varphi[/math].

There's a formula to calculate the n-th term of the sequence: [math]F_n=\frac{\left( 1+\sqrt{5} \right)^n-\left( 1- \sqrt{5} \right)^n }{2^n\sqrt{5}}[/math]. Splitting this formula you get [math]F_n=\frac{\Phi^n-\left(-\varphi\right)^n}{\sqrt{5}}[/math][br]The larger n, the smaller the second term in the denominator. Out of this one can deduce an approximate formula producing a better approximation for increasing values of n.

Notes: [br][list][*]for mathematicians: There's more mathematical background in the text [url=https://www.blogger.com/profile/08053866714451812324]Golden Maths & Myths[/url].[/*][*]for non-mathematicians: Just look at the approximate formula. It states:[br]There's a simple relation between the n-th term of the sequence of Fibonacci and [math]\Phi[/math].[br]In the next activity you can read what this means for the ''mysterious appearance of the golden ratio' in the sequence of Fibonacci.[/*][/list]