What is the Fibonacci Sequence (aka Fibonacci Series)?

Leonardo Fibonacci discovered the sequence which converges on phi.

Leonardo Fibonacci, discoverer of the Fibonacci series which is related to phi, the Golden Proportion In the 12th century, Leonardo Fibonacci wrote in Liber Abaci of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi.  This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa.

Starting with 0 and 1, each new number in the sequence 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 sequence 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 sequence quickly converge on Phi.  After the 40th number in the sequence, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .


Compute any number in the Fibonacci Sequence easily!

Here are two ways you can use phi to compute the nth number in the Fibonacci sequence (fn).

If you consider 0 in the Fibonacci sequence to correspond to n = 0, use this formula:

fn =  Phi n / 5½

Perhaps a better way is to consider 0 in the Fibonacci sequence 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
sequence
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

11
211.000000000000000+0.618033988749895
322.000000000000000-0.381966011250105
431.500000000000000+0.118033988749895
551.666666666666667-0.048632677916772
681.600000000000000+0.018033988749895
7131.625000000000000-0.006966011250105
8211.615384615384615+0.002649373365279
9341.619047619047619-0.001013630297724
10551.617647058823529+0.000386929926365
11891.618181818181818-0.000147829431923
121441.617977528089888+0.000056460660007
132331.618055555555556-0.000021566805661
143771.618025751072961+0.000008237676933
156101.618037135278515-0.000003146528620
169871.618032786885246+0.000001201864649
171,5971.618034447821682-0.000000459071787
182,5841.618033813400125+0.000000175349770
194,1811.618034055727554-0.000000066977659
206,7651.618033963166707+0.000000025583188
2110,9461.618033998521803-0.000000009771909
2217,7111.618033985017358+0.000000003732537
2328,6571.618033990175597-0.000000001425702
2446,3681.618033988205325+0.000000000544570
2575,0251.618033988957902-0.000000000208007
26121,3931.618033988670443+0.000000000079452
27196,4181.618033988780243-0.000000000030348
28317,8111.618033988738303+0.000000000011592
29514,2291.618033988754323-0.000000000004428
30832,0401.618033988748204+0.000000000001691
311,346,2691.618033988750541-0.000000000000646
322,178,3091.618033988749648+0.000000000000247
333,524,5781.618033988749989-0.000000000000094
345,702,8871.618033988749859+0.000000000000036
359,227,4651.618033988749909-0.000000000000014
3614,930,3521.618033988749890+0.000000000000005
3724,157,8171.618033988749897-0.000000000000002
3839,088,1691.618033988749894+0.000000000000001
3963,245,9861.618033988749895-0.000000000000000
40102,334,1551.618033988749895+0.000000000000000

Tawfik Mohammed notes that 13, considered by some to be an unlucky number, is found at position number 7, the lucky number!

The Fibonacci Sequence and Gambling or Lotteries

Some people hope that Fibonacci numbers will provide an edge in picking lottery numbers or bets in gambling. The truth is that the outcomes of games of chance are determined by random outcomes and have no special connection to Fibonacci numbers.

Roulette tables can use the Fibonacci method of betting There are, however, betting systems used to manage the way bets are placed, and the Fibonacci system based on the Fibonacci sequence is a variation on the Martingale progression. In this system, often used for casino and online roulette, the pattern of bets placed follows a Fibonacci progression: i.e., each wager should be the sum of the previous two wagers until a win is made. If a number wins, the bet goes back two numbers in the sequence because their sum was equal to the previous bet.

In the Fibonacci system the bets stay lower then a Martingale Progression, which doubles up every time. The downside is that in the Fibonacci roulette system the bet does not cover all of the losses in a bad streak.

An important caution: Betting systems do not alter the fundamental odds of a game, which are always in favor of the casino or the lottery. They may just be useful in making the playing of bets more methodical, as illustrated in the example below:

RoundScenario 1Scenario 2Scenario 3
Bet 1Bet 1 and loseBet 1 and loseBet 1 and win
Bet 2Bet 1 and loseBet 1 and loseBet 1 and win
Bet 3Bet 2 and winBet 2 and loseBet 1 and lose
Bet 4-Bet 3 and winBet 1 and lose
Bet 5--Bet 2 and win
Net ResultEven at 0Down by 1Ahead by 2

 

 

 

Comments

  1. matthew C Culver says

    FIBONACCI is the combinations of moves and or optimization
    one must make inorder to complete a task, taking in scenarios
    in which one would never lose.

  2. John says

    Thank you for your input and clarification sir. The original way is golden! You can never loose! Any other way can lead to a path of darkness and confusion as you try to come full circle.

    • says

      Thank you for the insight on this. There seem to be differing definitions depending on the source. Dictionary.com defines series as “a group or a number of related or similar things, events, etc., arranged or occurring in temporal, spatial, or other order or succession; sequence” followed by “Series, sequence, succession are terms for an orderly following of things one after another. Series is applied to a number of things of the same kind, usually related to each other, arranged or happening in order: a series of baseball games. Sequence stresses the continuity in time, thought, cause and effect, etc.: The scenes came in a definite sequence. Succession implies that one thing is followed by another or others in turn, usually though not necessarily with a relation or connection between them: succession to a throne; a succession of calamities.” Google lists 1.2 million references for “Fibonacci Series” and 2.1 million references for “Fibonacci sequence” so both are in common usage, although sequence is apparentely more prevalent. I’ll review your suggested changes and include these comments to the post for clarification.

      • says

        Gary – Very interesting article and table. FYI, Patrick is correct that series and sequence have specific meanings and are not interchangeable to mathematicians, no matter what Google or various dictionaries say. To mathematicians, a sequence is a progression of numbers generated by a function, whereas a series is the sum of numbers in a sequence. Your article is too good in other respects to use these terms in non-mathematical ways.
        Best,
        Lou

  3. Shelley says

    I am very curious about the “sequence” and how it affects us as people in our daily lives. John says it is the combinations of moves and or optimization one must make in order to complete a task, taking in scenarios in which one would never lose. Could you point me to more information how this connects with our lives, past, present and future? and if in laymen terms that would be even better.

    Thanks for your kind consideration of my request.

    Cheers Shelley

  4. ben says

    is the difference from phi column actually an inverted fibonacci series where you skip one number each time? 1+2=3, 2+3=5 but only 1,2 & 5 are in the sequence. next is 14, 36…

  5. Shivaji Results says

    I was looking for the real time application of Fibonacci Sequence and got it from your blog. Thank you Very Much for your awesome Article.

  6. William Vennard says

    One can begin with any two random numbers and as long as the Fibonacci pattern is followed, they will eventually come out to 1.6180339–!

    • says

      That is true. The Fibonacci numbers have some very unique properties of their own, however, and there’s something mathematically elegant to start with 0 and 1 rather than two randomly selected numbers. Either way, this illustrates the significance of the additive property of the Fibonacci series that allows us to derive phi from the ratios of the successive numbers.

    • pat says

      Adarsh, a “ratio” requires two things. Your question isn’t clear because you don’t say what two things you want the “ratio” of.

  7. pat says

    I noticed that there is actually an “exact” Fibonacci sequence. If you use phi (0.618…) as the first number and one as the second number, you get the sequence:

    0.6180339887, 1, 1.6180339887, 2.6180339887, 4.2360679775, 6.8541019662…

    I say it is “exact” because the ratio between successive terms is always exactly Phi (1.618…), with no approximation. This sequence has some interesting properties. The terms actually begin to approach integers as they get larger.

    • says

      The sequence of exponential powers of phi does have unique properties, but technically speaking it is not the sequence discovered by Fibonacci and named after him.

      • Ted says

        Hi Gary,

        If the Fibonacci sequence is the sequence starting with 1, what do we call the infinite number of other sequences whose ratios all converge on Phi in a similar manner?

        Any two starting numbers, including fractions or even negative numbers, in any combination, will work.

        Regards,

        Ted.

        • 12th Class Result 2014 says

          That depends on who invent the series. yes, there are many such series out there, but we need to identify them and need to prove their concept in front of the world. Publishing a paper on it will do the task.

  8. Result 2013 says

    To mathematicians, a sequence is a progression of numbers generated by a function, whereas a series is the sum of numbers in a sequence. Your article is too good in other respects to use these terms in non-mathematical ways.

    • Nick Fortis says

      Indeed. But a sequence need NOT be “generated by a function.” E.g.,
      2 6 13 8 1 41 (power ball choices, say), is a sequence. I may or may not wish to sum the sequence or form its product. Naf Saratoga CA ;-)

    • Nick Fortis says

      OK: again . USUALLY generated. The prime numbers form a sequence; One can surely determine them using various techniques, but no one can generate them.

      Unless you, perhaps, have solved RH. Or something related thereto.

      Naf

    • N A Fortis says

      Exception:
      “Random Sequence. A sequence that is irregular, non repetitive, and hapahazard. … …
      A completely satisfactory definition of randomn sequence is yet to be discovered. However, test of randomness can be made; e.g., by subdividing the sequence into blocks and using the chi-square test to to analyze the frequencies of occurrence of specified individual integers… … …A table of one million random digits has been published”

  9. Ted says

    I’ve also noticed that the ratio of successive pairs of numbers in other sum sequences converge as well. For example, take any three numbers and sum them to make a fourth, then continue summing the last three numbers in the sequence to make the next. The ratio of successive pairs of numbers in this sequence converges on 1.83928675521416….

    Similarly, summing the last four, five, six, seven and eight numbers converge on different values which themselves appear to converge on 2.0 as you increase the quantity of numbers which are summed. ie.;-

    Numbers Convergent value
    Summed

    2 1.61803398874989…
    3 1.83928675521416…
    4 1.92756197548293…
    5 1.96594823664549…
    6 1.98358284342433…
    7 1.99196419660503…
    8 1.99603117973541…

    Regards,
    Ted.

  10. zubaida says

    can someone tell me who the author of this article is? I would love to credit him or her for this wonderful job in my math project.

  11. N J Smith says

    Mr. Hawthorne’s comment is interesting, especially with respect to dictionary definitions.
    One sees that not all sequences can be generated by a function.

    The random sequence is one such (pg 247, Mathematics Dictionary, James & James, 5th Ed 1992.)

    “Random Sequence. A sequence that is irregular, non repetitive, and hapahazard. … …
    A completely satisfactory definition of random sequence is yet to be discovered. However, test of randomness can be made; e.g., by subdividing the sequence into blocks and using the chi-square test to to analyze the frequencies of occurrence of specified individual integers… … …A table of one million random digits has been published”

    Njs

  12. N J Smith says

    Mr. Hawthorne’s comment is interesting, especially with respect to dictionary definitions.
    One sees that not all sequences can be generated by a function.

    The random sequence is one such (pg 247, Mathematics Dictionary, James & James, 5th Ed 1992.
    “Random Sequence. A sequence that is irregular, non repetitive, and hapahazard. … …
    A completely satisfactory definition of randomn sequence is yet to be discovered. However, test of randomness can be made; e.g., by subdividing the sequence into blocks and using the chi-square test to to analyze the frequencies of occurrence of specified individual integers… … …A table of one million random digits has been published”

    Njs

  13. Dean Huffman says

    solved 432hz divided by 2 216,108,54, 27,13.5,6.75,3.375,1.6875 the atom inside a nucleus my head ,the one inside ,can see alot.

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