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					        Fibonacci Numbers
        Fibonacci numbers form a sequence. After two starting numbers, each additional number
        in the sequence is the sum of the two preceding numbers

        The first Fibonacci numbers are:
                0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,…..


        To create Fibonacci numbers using excel spreadsheets formulas, you need to select cells
        where you want to start the sequence, in this example, we are going to pick cell B21

        1. In cell B26 type a zero (0) and press the Enter key.
        2. In cell C26 type a 1 and press the Enter key.
        3. In cell D26 type the formula =B26+C26 and press the Enter key
        4. Click on cell D26 to copy the formula to the adjacent cells
        5. Place the mouse pointer over the fill handle in the bottom right corner of cell D26
        6. Hold down the mouse button on the fill handle and drag to the right to cell T26
        7. Cell T26 should contain the number 2584




0   1        1     2       3      5      8     13     21     34      55       89     144
ch additional number




need to select cells
to pick cell B21




rner of cell D26
ht to cell T26




                   233   377   610   987   1597   2584
Phi and the Fibonacci Series
Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical
relationship behind phi.
Starting with 0 and 1, each 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 succesive pair of numbers in the series approximates phi (1.618…), as 5 divided by 3 is 1.66…, and 8
divided by 5 is 1.60
Table A on next page shows how the ratios of succesive 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                                                                         Go to Table A
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:

                                                 Phi n
                                             fn  1 / 2
                                                  5
Pheraps 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 by Jordan Malachi Dant in April 2005

                                                       Phi n
                                             fn 
                                                    ( Phi  2)

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

The ratio of succesive Fibonacci numbers converges on Phi
Go to Table A
                                                      TABLE A

                               The ratio of succesive Fibonacci numbers converge

Sequence in the series   Resulting Fibonacci number     Ratio of each number
                                                         to the one before it
                  0                 0
                  1                 1
                  2                 1                     1.000000000000
                  3                 2                     2.000000000000
                  4                 3                     1.500000000000
                  5                 5                     1.666666666667
                  6                 8                     1.600000000000
                  7                13                     1.625000000000
                  8                21                     1.615384615385
                  9                34                     1.619047619048
                 10                55                     1.617647058824
                 11                89                     1.618181818182
                 12               144                     1.617977528090
                 13               233                     1.618055555556
                 14               377                     1.618025751073
                 15               610                     1.618037135279
                 16               987                     1.618032786885
                 17              1597                     1.618034447822
                 18              2584                     1.618033813400
                 19              4181                     1.618034055728
                 20              6765                     1.618033963167
numbers converges on Phi

           Difference from Phi




              0.61803309974989
             -0.38196690025011
              0.11803309974989
             -0.04863356691678
              0.01803309974989
             -0.00696690025011
              0.00264848436527
             -0.00101451929773
              0.00038604092636
             -0.00014871843193
              0.00005557166000
             -0.00002245580567
              0.00000734867693
             -0.00000403552862
              0.00000031286464
             -0.00000134807179
             -0.00000071365024
             -0.00000095597766
             -0.00000086341682
                                                               Fibonacci Numbers
    Fibonacci numbers form a sequence. .
    After two starting numbers, each additional number in the sequence is the sum of the two preceding numbers.
    The first Fibonacci numbers are:
                 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584 …


0            1          1            2        3            5        8         13         21         34            55   89   144




    Ratios of the Fibonacci Numbers
                         1       0.5 0.666667           0.6      0.625 0.615385 0.619048 0.617647 0.618182 0.617978 0.618056




        x                   Fib(x)                Ratios
        1                          1                       1
        2                          2                     0.5
        3                          3              0.666667
        4                          5                     0.6
        5                          8                  0.625
        6                         13              0.615385
        7                         21              0.619048
        8                         34              0.617647
        9                         55              0.618182
        10                        89              0.617978
        11                       144              0.618056
        12                       233              0.618026
13     377   0.618037
14     610   0.618033
15     987   0.618034
16    1597   0.618034
17    2584   0.618034
18    4181   0.618034
19    6765   0.618034
20   10946   0.618034
    233      377      610      987     1597      2584




0.618026 0.618037 0.618033 0.618034 0.618034 0.618034

				
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posted:11/20/2011
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