One of the more well know properties of the Fibonacci sequence is its relationship to the Golden Ratio (or Phi):

If you take the ratio of two consecutive
Fibonacci number (put the bigger number on top, and the smaller one on the
bottom) you will get a fraction that approximates the Golden Ratio. The larger the Fibonaccci numbers are, the
closer your approximation will be.

This is true, but what most people don’t
know is that this property is not unique to the Fibonacci sequence. If you start with numbers other than 1 and 1,
you will the sequence will still begin to approximate the value of the golden
ration closer and closer as the numbers get bigger. And you don’t have to pick integers – you can
choose fractions, decimals, or even negative numbers.

For example let’s look at the Lucas
sequence. The Lucas sequence begins with
2 and 1 (instead of 1 and 1), but otherwise works just like the Fibonacci
sequence.

The 41

^{st}Lucas number divided by the 40^{th}Lucas number is:
370248451 / 228826127 = 1.61803398874989480550007298773185983259682579865541...

The 41

^{st}Fibonacci number divided by the 40^{th}Fibonacci number is:
165580141 / 102334155 =

1.61803398874989489090910068099941803398874989489090...

1.61803398874989489090910068099941803398874989489090...

Both of these are the same up to 14 decimal
places! I don’t think I will ever need
to be more accurate, but If I do I can do it.

If I use the equation listed above for the
definition of the Golden Ration (Phi) is:

1.61803398874989484820458683436563811772030917980576...

So why did I pick the Lucas sequence? Well, the Lucas sequence hides another secret
about Phi that the Fibonacci sequence does not have. In the Lucas sequence the first two terms are
2 and 1. We will call them L

_{0}= 2, and L_{1}= 1. L_{2}= 3, which is equal to Phi^{2}rounded to the nearest integer. L_{3}= 4, which is Phi^{3}rounded to the nearest integer. In fact, this pattern continues to at least the 53^{rd}term. I suspect that it continues further, but my spreadsheet program is only accurate to 12 digits.
REFERENCES:

David

## No comments:

## Post a Comment