# Esssay

Topics: Fibonacci number, Fibonacci, Golden ratio Pages: 3 (741 words) Published: March 4, 2013
Compute any number in the Fibonacci Series easily!
Here are two ways you can use phi to compute the nth number in the Fibonacci series (fn). If you consider 0 in the Fibonacci series to correspond to n = 0, use this formula: fn = Phi n / 5½
Perhaps a better way is to consider 0 in the Fibonacci series to correspond to the 1st Fibonacci number where n = 1 for 0. Then you can use this formula, discovered and contributed by Jordan Malachi Dant in April 2005: fn = Phi n / (Phi + 2)

Both approaches represent limits which always round to the correct Fibonacci number and approach the actual Fibonacci number as n increases.

The ratio of successive Fibonacci numbers converges on phi
Sequence
in the
series| Resulting
Fibonacci
number
(the sum
of the two
numbers
before it)| Ratio of each
number to the
one before it
(this estimates
phi)| Difference
from
Phi|
|
0| 0| | |
1| 1| | |
2| 1| 1.000000000000000| +0.618033988749895|
3| 2| 2.000000000000000| -0.381966011250105|
4| 3| 1.500000000000000| +0.118033988749895|
5| 5| 1.666666666666667| -0.048632677916772|
6| 8| 1.600000000000000| +0.018033988749895|
7| 13| 1.625000000000000| -0.006966011250105|
8| 21| 1.615384615384615| +0.002649373365279|
9| 34| 1.619047619047619| -0.001013630297724|
10| 55| 1.617647058823529| +0.000386929926365|
11| 89| 1.618181818181818| -0.000147829431923|
12| 144| 1.617977528089888| +0.000056460660007|
13| 233| 1.618055555555556| -0.000021566805661|
14| 377| 1.618025751072961| +0.000008237676933|
15| 610| 1.618037135278515| -0.000003146528620|
16| 987| 1.618032786885246| +0.000001201864649|
17| 1,597| 1.618034447821682| -0.000000459071787|
18| 2,584| 1.618033813400125| +0.000000175349770|
19| 4,181| 1.618034055727554| -0.000000066977659|
20| 6,765| 1.618033963166707| +0.000000025583188|
21| 10,946| 1.618033998521803| -0.000000009771909|
22|...