Esssay
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 |
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 |