The approximate formula [math]F_n=\frac{\Phi^n}{\sqrt{5}}[/math] implies that approximately the sequence of Fibonacci is a geometrical sequence with starting value [math]\frac{1}{\sqrt{5}}[/math] and quotiënt [math]\Phi=1.618[/math].[br]Drawing this values upon the graph of [math]f\left(x\right)=\frac{\Phi^n}{\sqrt{5}}[/math] ore looking at the table of values, oen sees:[br][list][*]The numbers of the sequence of Fibonacci are in turn smaller and bigger than the approximate value.[/*][*]The difference gets smaller for increasing values of n. [/*][/list]In other words: [br][i]When you divide a number of the sequence of Fibonacci by the previous ons, than the quotient approximates the value 1.618... the further in the row, the better the approximation. [br][/i][i]So this value doesn't appear out of nothing if you come upon the idea of dividing terms. It's a consequence of the exact formula for the sequence of Fibonacci, in which already [/i][math]\varphi[/math][i] and [/i][math]\Phi[/math][i] appear.[br][/i]Some get euphoric is they see en random number from the sequence of Fibonacci en believe they have spotted the golden ratio. Doing so is mixing apples and pears. 2, 3, 5 and 8 are very current Fibonacci . In nature even larger Fibonacci numbers appear. You can read this in the activity about sunflowers. However [math]\Phi[/math] is a limit value, when n becomes infinitly big. That's why Fibonacci numbers occur in nature, but [math]\Phi[/math] doesn't.

Chris Impens concludes: [br][i]"Fibonacci numbers occur in physical reality and can be undoubtly counted. [br]Irrational numbers cannot be counted. [br]According to [url=https://en.wikipedia.org/wiki/Occam%27s_razor]Occam's razor[/url] the first option (Fibonacci number) is to be preferred .[br]In other words: Fibonacci numbers are real, while the golden number is a mathematical artefact, arising from idealization."[/i]