exact and approximate Fibonacci formula
the golden ratio and the sequence of Fibonacci
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 .
Then again, where does this link come from?
Whell, likewise 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 , you can even rewrite the formula using and .
a formula for the sequence of Fibonacci
There's a formula to calculate the n-th term of the sequence: . Splitting this formula you get
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:
- for mathematicians: There's more mathematical background in the text Golden Maths & Myths.
- for non-mathematicians: Just look at the approximate formula. It states: There's a simple relation between the n-th term of the sequence of Fibonacci and . In the next activity you can read what this means for the ''mysterious appearance of the golden ratio' in the sequence of Fibonacci.