## Leonardo Fibonacci discovered the sequence which converges on phi.

In the 1202 AD, Leonardo Fibonacci wrote in his book “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. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning “son of (the) Bonacci”.

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, 233, 377 . . .

This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze. Click to enlarge.

The relationship of the Fibonacci sequence to the golden ratio is this: 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 (f_{n}).

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

f_{n} = Phi^{ n} / 5^{½}

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:

f_{n} = 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 | ||

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, 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.

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:

Round | Scenario 1 | Scenario 2 | Scenario 3 |

Bet 1 | Bet 1 and lose | Bet 1 and lose | Bet 1 and win |

Bet 2 | Bet 1 and lose | Bet 1 and lose | Bet 1 and win |

Bet 3 | Bet 2 and win | Bet 2 and lose | Bet 1 and lose |

Bet 4 | – | Bet 3 and win | Bet 1 and lose |

Bet 5 | – | – | Bet 2 and win |

Net Result | Even at 0 | Down by 1 | Ahead by 2 |

matthew C Culver says

June 11, 2012 at 9:06 pmDANTS FORMULA IS THE LOG OF ONE DEFINED DIMENSION TO THE DIVISION OF ITSELF

Matt says

July 28, 2014 at 11:28 pmI 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 much better.

matthew C Culver says

June 11, 2012 at 9:10 pmFIBONACCI 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.

John says

July 30, 2012 at 7:07 amThank 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.

Patrick says

September 12, 2012 at 8:45 pmI love the column, but it hits something of a pet peeve. Check out

http://en.wikipedia.org/wiki/Series_(mathematics)

to see the distinction between a sequence and a series. Basically, everywhere you see the word “series”, it should be “sequence”. Instead of “Sequence in the series”, how about “Position in the sequence”.

Gary Meisner says

September 24, 2012 at 7:51 pmThank 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.

Lou Hawthorne says

March 28, 2013 at 3:11 pmGary – 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

Gary Meisner says

March 29, 2013 at 10:24 pmThanks, Lou. I’ve taken your advice and changed the references in the article to sequence from series.

nick fortis says

February 24, 2014 at 4:24 pmAnd a product as well.

Shelley says

November 5, 2012 at 3:56 amI 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

lucas says

March 18, 2016 at 8:09 ami am to0.

lucas says

March 18, 2016 at 8:12 amHe must have been absolutely amazing figuring this out without calculators.

ben says

December 24, 2012 at 6:36 amis 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…

Janne says

January 9, 2013 at 4:36 amThank you …

Martin says

March 31, 2013 at 8:35 pmHow brilliant he must have been. And now we use calculators. Thanks — Martin

Mike E. says

April 5, 2013 at 11:49 amNor sure if you’ve seen the work done by artist Vi Hart posted on Kahn Academy. If not, enjoy. Love your site.

https://www.khanacademy.org/math/recreational-math/vi-hart

hardik says

May 13, 2013 at 8:31 amawesome!!!!!!!!!!!!!!!!!!!!!!!!!!

HHHProgram says

June 5, 2013 at 2:02 amHey Gary Meisner,

Excellent article for the Fibonacci series of course this blog is doing a very good job of serving useful information. I’m proud to be a part of its Readers community.

For the Fibonacci programs in different languange like C language,JAVA,C# must visit http://www.hhhprogram.com/2013/05/fibonaccci-series.html

Results 2013 says

July 23, 2013 at 11:54 pmThank you hardik for your Good job.

stephen says

July 16, 2013 at 3:04 pmIs it posible that Fibonaccis Sequence could explane the bigbang or how time started????

Eros says

October 16, 2013 at 11:25 pmYes, the big bang was the result of the Golden Number being divided by zero. So, never do that!

Eros.

Gary Meisner says

October 18, 2013 at 4:46 amGood humor. Division by zero is known to mess up calculators and spreadsheets, but current thinking in cosmology reflects a different cause. There is an interesting relationship though between 0 divided by 1 and Phi discussed on Theology page.

David f says

December 20, 2014 at 1:26 pmhttps://groups.google.com/d/msg/sci.physics.relativity/EHtG-Zz33_Q/zcSOIzVAQA8J

http://physics.nist.gov/cgi-bin/cuu/Value?mu0%7Csearch_for=universal_in!

4pi x 2 x 3 x 5

0.99930819635258

Value 376.730 313 461… Omega

Standard uncertainty (exact)

Relative standard uncertainty (exact)

Concise form 376.730 313 461… Omega

(376.7303146)/(4*π*2*3*5)*300000000

299792458.90577

Mike. says

October 15, 2016 at 7:03 amSpirit science talks alot of this subject. I believe its called sacred geometry. It shows alot of the ways phi and fibonaci occur EVERYWHERE in the universe. From conch shells to DNA, to expanding galaxies! The whole series is very informative, a new perspective of seeing the things we see constantly.

Gary B Meisner says

October 16, 2016 at 10:56 pm“EVERYWHERE” is not completely accurate. It appears many places, but many spirals in nature are just equiangular spirals and not golden spirals.

Shivaji Results says

July 18, 2013 at 2:25 pmI was looking for the real time application of Fibonacci Sequence and got it from your blog. Thank you Very Much for your awesome Article.

William Vennard says

August 31, 2013 at 11:57 amOne can begin with any two random numbers and as long as the Fibonacci pattern is followed, they will eventually come out to 1.6180339–!

Gary Meisner says

August 31, 2013 at 2:35 pmThat 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.

AmberEliana says

March 1, 2014 at 1:13 pmHowever, Fibonacci sequence converges faster than other similar sequences.

Adarsh nimje says

September 11, 2013 at 12:14 pmCAN ANYONE TELL ME WHAT IS THE RATIO OF AN ANGLE OF GOLDEN TRIANGLE???????

pat says

September 28, 2013 at 2:54 pmAdarsh, a “ratio” requires two things. Your question isn’t clear because you don’t say what two things you want the “ratio” of.

pat says

September 28, 2013 at 2:51 pmI 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.

Gary Meisner says

October 1, 2013 at 3:17 amThe 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

October 16, 2013 at 12:15 amHi 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.

Ted says

October 16, 2013 at 12:23 amSorry, misspelled Fibonacci!

Ted.

12th Class Result 2014 says

March 11, 2014 at 1:00 amThat 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.

tristan says

October 4, 2013 at 7:42 pmthat’s pretty easy

tina says

October 7, 2013 at 6:28 amthanks for helping :))))))))))))))))))))))

Result 2013 says

November 8, 2013 at 11:28 pmTo 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

February 25, 2014 at 9:43 amIndeed. 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

February 25, 2014 at 4:12 pmOK: 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

February 25, 2014 at 7:50 pmException:

“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”

Ted says

November 12, 2013 at 11:31 pmI’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.

Ted says

November 18, 2013 at 8:18 pmI guess I should have Googled this earlier;-

http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers

Regards,

Ted.

Mahesh chandra says

January 15, 2014 at 7:55 amI do appreciate and you have done nice work!! and i always use http://en.wikipedia.org/wiki/Series_(mathematics), gives more information

zubaida says

January 22, 2014 at 2:49 pmcan 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.

Gary Meisner says

January 23, 2014 at 9:02 pmUnless otherwise noted, all articles on this site are written by Gary Meisner. See https://www.goldennumber.net/content-images-use for details on references.

N J Smith says

February 25, 2014 at 6:29 pmMr. 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

N J Smith says

February 25, 2014 at 6:31 pmMr. 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

harsh says

March 7, 2014 at 4:04 amHow brilliant he must have been. And now we use calculators. Thanks for this informative article.

Dean Huffman says

March 17, 2014 at 12:39 pmsolved 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.

Gaurav says

May 6, 2014 at 8:37 amNow a days we use calculators….How brilliant he must have been.

deepjyoti deb says

August 24, 2014 at 11:18 pmcan u pls tell me dat which Indian or in which Indian book phi is discovered 1st

JanetAlexander says

September 27, 2014 at 12:55 pmCorrect pronunciation is FEE

Gary Meisner says

September 28, 2014 at 11:41 pmLike many other words in the English language, the answer depends on who you ask and where you ask it. See https://www.goldennumber.net/pronouncing-phi/ for a more in depth discussion.

Wesley Horton says

October 21, 2014 at 12:31 pmQuestion:

If you pick a random number N (lets say 17) and N+1 (18) and started the sequence from those two numbers, does the series converge on phi or some other infinite series?

Gary Meisner says

October 21, 2014 at 9:14 pmYou can start with any two numbers, add then together and continue in the same way and the ratio of the larger to the smaller will converge on phi.

NMK says

May 30, 2016 at 11:54 amThe 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

Richard Allan Kretzschmar II/Jr says

November 24, 2014 at 1:07 amThere is even more to this brilliance of the (phi) magic as the synchronous nature of the letters PHI serve us as a mnemonic acronym for the languages (Polish-Haitian-Igbo) as the New World Order of the Northwest manifest a spoken “Golden Motto” phrase from the well of the almighty Torus; a surface of revolution generated by revolving a circle in three-dimensional space throat. In military quantum theology theory this is equivalent to the word of God and or All other collective deities of the X, Y, and Z axis. RAK II

Remi says

January 15, 2015 at 12:05 amA Fibo fact I want to share:

ONE+ONE+TWO+THREE+FIVE+EIGHT = 13×21

The sum of gematrias of the 6 first Fibos gives the product of the 2 next terms with an incredible reciprocity:

1x1x2x3x5x8 = THIRTEEN+TWENTYONE

The product of the 6 first Fibos gives the sum of gematrias of the 2 next terms

See the verification here http://www.gef.free.fr/gem.php?texte=ONE+ONE+TWO+THREE+FIVE+EIGHTTHIRTEEN+TWENTYONE

Gary Meisner says

January 15, 2015 at 10:34 amThat’s a rather amazing intersection of numbers and letters. For those who aren’t familiar with “gematria” it simply means in this case assigning a number value to each letter. A = 1, B = 2, C = 3, D = 4, etc. This works for the Fibonacci numbers in English. What about other languages?

IGNOU says

May 15, 2015 at 5:43 amThanks for this equation.

Solar Fence says

July 13, 2015 at 6:17 amFibonacci sequence converges faster than other similar sequences. Plus, you can start it with any two numbers.

Peter Hedding says

August 27, 2015 at 9:09 pmI’m no mathematician or scientist, but from what I understand about bra-ket notation, just about everything grows and then decays according to logarithmic spirals and whirling squares, represented by PSI and PHI.

And what I’ve read seems to say that there are other possible logarithmic spirals the universe could be based

on, but the PHI spiral is the slowest of all, and using any of the others would have made it impossible for life on earth to exist at all! Any expert opinions out there to shed more light on this notion?

RB says

February 24, 2016 at 12:29 amHeard of Hypatia?

Daniel says

May 26, 2017 at 9:29 pmYup… great female thinker and scientist of her time in Egypt. Sadly condemned by those ‘pious’, ‘self-righteous’ and intolerant ignoramii Christians of her time. First for being an outspoken woman and second for defying normal conventions and her intelligence.

Garry says

March 6, 2017 at 9:39 amTo Peter Hedding

Please tell me more about bra-ket notion! It’s very important to me. I want to use in a lottery game. If it possible for you I think it’s gonna be okay to describe more than one lottery strategies. Perhaps you help me to win this lottery:

http://australian-lotto-results.com/ozlotto

Thanks!

Rajesh Shrestha says

April 12, 2017 at 12:22 pmHowever, this mathematical sequence has been already descrived in Vedas and long later By Aryabhatta and Bhaskar- the great scholars of Vedic culture of Nepal. http://www.tushitanepal.com

Daniel says

May 26, 2017 at 9:25 pmI first became interested in the Fibonacci sequence when I asked one of my high school science teachers how he explained that curls of hair and desert sand dunes seen from above seem to have the same pattern. He mentioned Fibonacci and Pascal and I was hooked. But the picture that stands out most as a Fibonacci reminder is that of a green vegetable resembling a broccoli.

Gary B Meisner says

May 30, 2017 at 11:03 amFibonacci number patterns do appear in nature, but be careful in using them as an explanation. Most curves and spirals in nature, particularly in non-living examples, are simply equiangular / logarhymic curves, which expand at an equal pace throughout the curve and have nothing to do with Fibonacci numbers or the golden ratio.

Klaas says

July 5, 2018 at 4:44 pmIt’s called a Romanesco broccoli..

Ando says

March 18, 2018 at 5:38 amThe so-called Fibonacci set was actually discovered by the ancient Indian mathematician Pingala in the 2nd or 3rd century BCE (the same guy who discovered binary system).

Anonymous says

December 1, 2018 at 5:38 amCan someone tell me WHY fibonacci thiught it was interesting

Aritina says

April 14, 2019 at 2:01 amI wonder if one could use this function to predict human history based on past prectable behaviors to certain social/historical/psychological stimuli- kinda like psychohistory in Asimov’s Foundation series.