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Phi and the Fibonacci Series
Leonardo Fibonacci discovered the series which converges on
phi
 In
the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the
foundation for an incredible mathematical relationship behind phi.
Starting with 0 and 1, each new number in the series is simply the sum of
the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
The ratio of each successive pair of numbers in the series approximates
phi (1.618. . .) , as 5 divided by
3 is 1.666..., and 8 divided by 5 is 1.60.
The table below shows how the ratios of the successive numbers in the
Fibonacci series quickly converge on Phi. After the 40th number in the series,
the ratio is accurate to 15 decimal places.
1.618033988749895 . . .
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 |
17,711 |
1.618033985017358 |
+0.000000003732537 |
| 23 |
28,657 |
1.618033990175597 |
-0.000000001425702 |
| 24 |
46,368 |
1.618033988205325 |
+0.000000000544570 |
| 25 |
75,025 |
1.618033988957902 |
-0.000000000208007 |
| 26 |
121,393 |
1.618033988670443 |
+0.000000000079452 |
| 27 |
196,418 |
1.618033988780243 |
-0.000000000030348 |
| 28 |
317,811 |
1.618033988738303 |
+0.000000000011592 |
| 29 |
514,229 |
1.618033988754323 |
-0.000000000004428 |
| 30 |
832,040 |
1.618033988748204 |
+0.000000000001691 |
| 31 |
1,346,269 |
1.618033988750541 |
-0.000000000000646 |
| 32 |
2,178,309 |
1.618033988749648 |
+0.000000000000247 |
| 33 |
3,524,578 |
1.618033988749989 |
-0.000000000000094 |
| 34 |
5,702,887 |
1.618033988749859 |
+0.000000000000036 |
| 35 |
9,227,465 |
1.618033988749909 |
-0.000000000000014 |
| 36 |
14,930,352 |
1.618033988749890 |
+0.000000000000005 |
| 37 |
24,157,817 |
1.618033988749897 |
-0.000000000000002 |
| 38 |
39,088,169 |
1.618033988749894 |
+0.000000000000001 |
| 39 |
63,245,986 |
1.618033988749895 |
-0.000000000000000 |
| 40 |
102,334,155 |
1.618033988749895 |
+0.000000000000000 |
Tawfik Mohammed notes
that 13, the unlucky number, is found at position number 7, the lucky
number! |
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Investors:
Apply
Phi and
Fibonacci
principles
to the
stock market |
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