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  • Topic: Fibonacci number, Fibonacci, Golden ratio
  • Pages : 3 (741 words )
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  • Published : March 4, 2013
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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|...
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