arly as the 6th century ad by indian mathematicians

8/16/2019 Arly as the 6th Century AD by Indian Mathematicians

1/42

arly 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 Af You are here: Home / Phi / What is the

Fibonacci Sequence (aka Fibonacci Series)?

What is the Fibonacci Seuence !a"aFibonacci Series#$May 15 !"1! by #ary Meisner $% &omments

%eonardo Fibonacci discovered the seuencewhich converges on &hi'

'n the 1!th century eonaro Fibonacci *rote in iber +baci o, a sim-.e numerica. sequencethat is the ,ounation ,or an increib.e mathematica. re.ationshi- behin -hi 0his sequence *as

kno*n as ear.y as the $th century + by 'nian mathematicians but it *as Fibonacci *ho

introuce it to the *est a,ter his tra2e.s throu3hout the Meiterranean *or. an

4orth +,rica

Startin3 *ith " an 1 each ne* number in the sequence is sim-.y the sum o, the

t*o be,ore it

" 1 1 ! % 5 1% !1 %6 55 7 166

0he ratio o, each successi2e -air o, numbers in the sequence a--ro8imates -hi (1$1 )

as 5 i2ie by % is 1$$$9 an i2ie by 5 is 1$"

0he tab.e be.o* sho*s ho* the ratios o, the successi2e numbers in the Fibonacci sequence

quick.y con2er3e on Phi +,ter the 6"th number in the sequence the ratio is accurate to 15

ecima. -.aces

('6()*++))-.)/ ' ' '

http://www.goldennumber.net/http://www.goldennumber.net/category/phi-basics/http://www.goldennumber.net/author/phipoint/http://www.goldennumber.net/author/phipoint/http://www.goldennumber.net/author/phipoint/http://www.goldennumber.net/fibonacci-series/#commentshttp://www.goldennumber.net/category/phi-basics/http://www.goldennumber.net/author/phipoint/http://www.goldennumber.net/fibonacci-series/#commentshttp://www.goldennumber.net/


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" "

1 1

! 1 1""""""""""""""" ="$1"%%7@6775

% ! !""""""""""""""" A"%17$$"11!5"1"5

6 % 15"""""""""""""" ="11"%%7@6775

5 5 1$$$$$$$$$$$$$$@ A""6$%!$@@71$@@!

$ 1$"""""""""""""" =""1"%%7@6775

@ 1% 1$!5"""""""""""" A"""$7$$"11!5"1"5

!1 1$15%6$15%6$15 ="""!$67%@%%$5!@7

7 %6 1$17"6@$17"6@$17 A"""1"1%$%"!7@@!6

1" 55 1$1@$6@"5!%5!7 =""""%$7!77!$%$5

11 7 1$1111111 A""""16@!76%17!%

1! 166 1$1@7@@5!"7 ="""""5$6$"$$"""@

1% !%% 1$1"5555555555$ A"""""!15$$"5$$1

16 %@@ 1$1"!5@51"@!7$1 =""""""!%@$@$7%%

15 $1" 1$1"%@1%5!@515 A""""""%16$5!$!"

1$ 7@ 1$1"%!@$5!6$ =""""""1!"1$6$67

1@ 157@ 1$1"%666@!1$! A"""""""657"@1@@


4/42

1 !56 1$1"%%1%6""1!5 ="""""""1@5%67@@"

17 611 1$1"%6"55@!@556 A""""""""$$7@@$57

!" $@$5 1$1"%%7$%1$$@"@ =""""""""!55%1

!1 1"76$ 1$1"%%775!1"% A"""""""""7@@17"7

!! 1@@11 1$1"%%75"1@%5 ="""""""""%@%!5%@

!% !$5@ 1$1"%%77"1@557@ A"""""""""16!5@"!

!6 6$%$ 1$1"%%7!"5%!5 =""""""""""5665@"

!5 @5"!5 1$1"%%775@7"! A""""""""""!"""@

!$ 1!1%7% 1$1"%%7$@"66% ="""""""""""@765!

!@ 17$61 1$1"%%7@"!6% A"""""""""""%"%6

! %1@11 1$1"%%7@%%"% ="""""""""""1157!

!7 516!!7 1$1"%%7@56%!% A""""""""""""66!

%" %!"6" 1$1"%%7@6!"6 =""""""""""""1$71

%1 1%6$!$7 1$1"%%7@5"561 A"""""""""""""$6$

%! !1@%"7 1$1"%%7@67$6 ="""""""""""""!6@

%% %5!65@ 1$1"%%7@6777 A""""""""""""""76

%6 5@"!@ 1$1"%%7@6757 =""""""""""""""%$

%5 7!!@6$5 1$1"%%7@677"7 A""""""""""""""16

%$ 167%"%5! 1$1"%%7@677" ="""""""""""""""5

%@ !615@1@ 1$1"%%7@677@ A"""""""""""""""!

% %7"1$7 1$1"%%7@6776 ="""""""""""""""1


5/42

%7 $%!657$ 1$1"%%7@6775 A""""""""""""""""

6" 1"!%%6155 1$1"%%7@6775 =""""""""""""""""

0a*,ik Mohamme notes that 1% consiere by some to be an un.ucky number is ,oun at

-osition number @ the .ucky numberB

2he Fibonacci Seuence and 3ambling or%otteries

Some -eo-.e ho-e that Fibonacci numbers *i.. -ro2ie an e3e in -ickin3 .ottery numbers or bets in 3amb.in3 0he truth is that the outcomes o, 3ames o, chance are

etermine by ranom outcomes an ha2e no s-ecia. connection to

Fibonacci numbers

0here are ho*e2er bettin3 systems use to mana3e the *ay bets are

-.ace an the Fibonacci system base on the Fibonacci sequence is a2ariation on the Martin3a.e -ro3ression 'n this system o,ten use ,or

casino an on.ine rou.ette the -attern o, bets -.ace ,o..o*s a Fibonacci

-ro3ression: ie each *a3er shou. be the sum o, the -re2ious t*o

*a3ers unti. a *in is mae ', a number *ins the bet 3oes back t*o numbers in the sequence because their sum *as equa. to the -re2ious bet

'n the Fibonacci system the bets stay .o*er then a Martin3a.e Pro3ression *hich

oub.es u- e2ery time 0he o*nsie is that in the Fibonacci rou.ette system the betoes not co2er a.. o, the .osses in a ba streak

+n im-ortant caution: >ettin3 systems o not a.ter the ,unamenta. os o, a 3ame

*hich are a.*ays in ,a2or o, the casino or the .ottery 0hey may Cust be use,u. in

makin3 the -.ayin3 o, bets more methoica. as i..ustrate in the e8am-.e be.o*:

Scenario %

>et 1 an *in

>et 1 an *in


6/42

>et ! an .ose

>et % an *in

D

o*n by 1

Fi.e Ener: Phi

0omments

1 matthe* & &u.2er says

http://www.goldennumber.net/category/phi-basics/http://www.goldennumber.net/category/phi-basics/


7/42


8/42


9/42


10/42

ou Ha*thorne says

March ! !"1% at %:11 -m

#ary D ery interestin3 artic.e an tab.e FY' Patrick is correct that series an sequence ha2e s-eci,ic

meanin3s an are not interchan3eab.e to mathematicians no matter *hat #oo3.e or 2arious ictionariessay 0o mathematicians a sequence is a -ro3ression o, numbers 3enerate by a ,unction *hereas a serie

the sum o, numbers in a sequence Your artic.e is too 3oo in other res-ects to use these terms in nonA

mathematica. *ays

>estou

e-.y

#ary Meisner says

March !7 !"1% at 1":!6 -m

0hanks ou 'N2e taken your a2ice an chan3e the re,erences in the artic.e to sequence ,rom series

e-.y

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11/42

nick ,ortis says

February !6 !"16 at 6:!6 -m

+n a -rouct as *e..

e-.y

5 She..ey says

4o2ember 5 !"1! at %:5$ am

' am 2ery curious about the IsequenceJ an ho* it a,,ects us as -eo-.e in our ai.y .i2es


12/42

e-.y

o .ucas says

March 1 !"1$ at :"7 am

i am to"

e-.y

.ucas says

March 1 !"1$ at :1! am

He must ha2e been abso.ute.y amaKin3 ,i3urin3 this out *ithout ca.cu.ators

e-.y

http://www.goldennumber.net/fibonacci-series/?replytocom=601#respondhttp://www.goldennumber.net/fibonacci-series/#comment-2901http://www.goldennumber.net/fibonacci-series/?replytocom=2901#respondhttp://www.goldennumber.net/fibonacci-series/#comment-2902http://www.goldennumber.net/fibonacci-series/?replytocom=2902#respondhttp://www.goldennumber.net/fibonacci-series/?replytocom=601#respondhttp://www.goldennumber.net/fibonacci-series/#comment-2901http://www.goldennumber.net/fibonacci-series/?replytocom=2901#respondhttp://www.goldennumber.net/fibonacci-series/#comment-2902http://www.goldennumber.net/fibonacci-series/?replytocom=2902#respond


13/42


14/42

Martin says

March %1 !"1% at :%5 -m

Ho* bri..iant he must ha2e been +n no* *e use ca.cu.ators 0hanks Martin

e-.y

7 Mike G says

+-ri. 5 !"1% at 11:67 am

4or sure i, youN2e seen the *ork one by artist i Hart -oste on Qahn +caemy ', not enCoy

o2e your site

htt-s://***khanacaemyor3/math/recreationa.Amath/2iAhart

e-.y

http://calendartoprint.com/http://www.goldennumber.net/fibonacci-series/#comment-613http://www.goldennumber.net/fibonacci-series/?replytocom=613#respondhttps://www.khanacademy.org/math/recreational-math/vi-harthttp://www.goldennumber.net/fibonacci-series/#comment-614https://www.khanacademy.org/math/recreational-math/vi-harthttp://www.goldennumber.net/fibonacci-series/?replytocom=614#respondhttp://calendartoprint.com/http://www.goldennumber.net/fibonacci-series/#comment-613http://www.goldennumber.net/fibonacci-series/?replytocom=613#respondhttps://www.khanacademy.org/math/recreational-math/vi-harthttp://www.goldennumber.net/fibonacci-series/#comment-614https://www.khanacademy.org/math/recreational-math/vi-harthttp://www.goldennumber.net/fibonacci-series/?replytocom=614#respond


15/42

1" harik says

May 1% !"1% at :%1 am

a*esomeBBBBBBBBBBBBBBBBBBBBBBBBBB

e-.y

11 HHHPro3ram says


16/42

o esu.ts !"1% says


17/42

o Gros says

ctober 1$ !"1% at 11:!5 -m

Yes the bi3 ban3 *as the resu.t o, the #o.en 4umber bein3 i2ie by Kero So ne2er o thatB

Gros

e-.y

#ary Meisner says

ctober 1 !"1% at 6:6$ am

#oo humor i2ision by Kero is kno*n to mess u- ca.cu.ators an s-reasheets but current thinkin3 in

cosmo.o3y re,.ects a i,,erent cause 0here is an interestin3 re.ationshi- thou3h bet*een " i2ie by 1

an Phi iscusse on 0heo.o3y -a3e

e-.y

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18/42

o a2i , says

ecember !" !"16 at 1:!$ -m

htt-s://3rou-s3oo3.ecom//ms3/sci-hysicsre.ati2ity/GHt#AK%%LT/KcS'K+T+<

htt-://-hysicsnist3o2/c3iAbin/cuu/a.ue?mu"U@&searchL,oruni2ersa.LinB

6-i 8 ! 8 % 8 5

"777%"17$%5!5

a.ue %@$@%" %1% 6$19 me3a

Stanar uncertainty (e8act)

e.ati2e stanar uncertainty (e8act)

&oncise ,orm %@$@%" %1% 6$19 me3a

(%@$@%"%16$)/(6VV!V%V5)V%""""""""

!77@7!657"5@@

e-.y

http://www.goldennumber.net/fibonacci-series/#comment-2031https://groups.google.com/d/msg/sci.physics.relativity/EHtG-Zz33_Q/zcSOIzVAQA8Jhttp://physics.nist.gov/cgi-bin/cuu/Value?mu0%7Csearch_for=universal_inhttp://www.goldennumber.net/fibonacci-series/?replytocom=2031#respondhttp://www.goldennumber.net/fibonacci-series/#comment-2031https://groups.google.com/d/msg/sci.physics.relativity/EHtG-Zz33_Q/zcSOIzVAQA8Jhttp://physics.nist.gov/cgi-bin/cuu/Value?mu0%7Csearch_for=universal_inhttp://www.goldennumber.net/fibonacci-series/?replytocom=2031#respond


19/42


20/42

o #ary Meisner says

+u3ust %1 !"1% at !:%5 -m

0hat is true 0he Fibonacci numbers ha2e some 2ery unique -ro-erties o, their o*n ho*e2er an

thereNs somethin3 mathematica..y e.e3ant to start *ith " an 1 rather than t*o ranom.y se.ecte

numbers Gither *ay this i..ustrates the si3ni,icance o, the aiti2e -ro-erty o, the Fibonacci seriesthat a..o*s us to eri2e -hi ,rom the ratios o, the successi2e numbers

e-.y

o +mberG.iana says

March 1 !"16 at 1:1% -m

Ho*e2er Fibonacci sequence con2er3es ,aster than other simi.ar sequences

e-.y

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21/42

15 +arsh nimCe says

Se-tember 11 !"1% at 1!:16 -m

&+4 +4Y4G 0G MG WH+0 'S 0HG +0' F +4 +4#G F #G4

0'+4#G???????

e-.y

o -at says

Se-tember ! !"1% at !:56 -m

+arsh a IratioJ requires t*o thin3s Your question isnNt c.ear because you onNt say *hat t*o thin3s

you *ant the IratioJ o,

e-.y

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22/42

1$ -at says

Se-tember ! !"1% at !:51 -m

' notice that there is actua..y an Ie8actJ Fibonacci sequence ', you use -hi ("$19) as the

,irst number an one as the secon number you 3et the sequence:

"$1"%%7@ 1 1$1"%%7@ !$1"%%7@ 6!%$"$@7@@5 $561"17$$!9

' say it is Ie8actJ because the ratio bet*een successi2e terms is a.*ays e8act.y Phi (1$19)

*ith no a--ro8imation 0his sequence has some interestin3 -ro-erties 0he terms actua..y be3in

to a--roach inte3ers as they 3et .ar3er

e-.y

o #ary Meisner says

ctober 1 !"1% at %:1@ am

0he sequence o, e8-onentia. -o*ers o, -hi oes ha2e unique -ro-erties but technica..y s-eakin3 it is

not the sequence isco2ere by Fibonacci an name a,ter him

http://www.goldennumber.net/fibonacci-series/#comment-630http://www.goldennumber.net/fibonacci-series/?replytocom=630#respondhttp://www.goldennumber.net/http://www.goldennumber.net/fibonacci-series/#comment-632http://www.goldennumber.net/fibonacci-series/#comment-630http://www.goldennumber.net/fibonacci-series/?replytocom=630#respondhttp://www.goldennumber.net/http://www.goldennumber.net/fibonacci-series/#comment-632


23/42

e-.y

0e says

ctober 1$ !"1% at 1!:15 am

Hi #ary

', the Fibonacci sequence is the sequence startin3 *ith 1 *hat o *e ca.. the in,inite number o, othersequences *hose ratios a.. con2er3e on Phi in a simi.ar manner?

+ny t*o startin3 numbers inc.uin3 ,ractions or e2en ne3ati2e numbers in any combination *i.. *ork

e3ars

0e

e-.y

0e says

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24/42

ctober 1$ !"1% at 1!:!% am

Sorry miss-e..e FibonacciB

0e

e-.y

1!th &.ass esu.t !"16 says

March 11 !"16 at 1:"" am

0hat e-ens on *ho in2ent the series yes there are many such series out there but *e nee to ienti,y

an nee to -ro2e their conce-t in ,ront o, the *or. Pub.ishin3 a -a-er on it *i.. o the task

e-.y

1@ tristan says

http://www.goldennumber.net/fibonacci-series/#comment-636http://www.goldennumber.net/fibonacci-series/?replytocom=636#respondhttp://www.goldennumber.net/fibonacci-series/#comment-660http://www.goldennumber.net/fibonacci-series/?replytocom=660#respondhttp://www.goldennumber.net/fibonacci-series/#comment-636http://www.goldennumber.net/fibonacci-series/?replytocom=636#respondhttp://www.goldennumber.net/fibonacci-series/#comment-660http://www.goldennumber.net/fibonacci-series/?replytocom=660#respond


25/42

ctober 6 !"1% at @:6! -m

thatNs -retty easy

e-.y

o tina says

ctober @ !"1% at $:! am

thanks ,or he.-in3 :))))))))))))))))))))))

e-.y

1 esu.t !"1% says

4o2ember !"1% at 11:! -m

http://www.goldennumber.net/fibonacci-series/#comment-633http://www.goldennumber.net/fibonacci-series/?replytocom=633#respondhttp://www.goldennumber.net/fibonacci-series/#comment-634http://www.goldennumber.net/fibonacci-series/?replytocom=634#respondhttp://www.goldennumber.net/fibonacci-series/#comment-640http://www.goldennumber.net/fibonacci-series/#comment-633http://www.goldennumber.net/fibonacci-series/?replytocom=633#respondhttp://www.goldennumber.net/fibonacci-series/#comment-634http://www.goldennumber.net/fibonacci-series/?replytocom=634#respondhttp://www.goldennumber.net/fibonacci-series/#comment-640


26/42

0o mathematicians a sequence is a -ro3ression o, numbers 3enerate by a ,unction *hereas a

series is the sum o, numbers in a sequence Your artic.e is too 3oo in other res-ects to use these

terms in nonAmathematica. *ays

e-.y

o 4ick Fortis says

February !5 !"16 at 7:6% am

'nee >ut a sequence nee 40 be I3enerate by a ,unctionJ G3

! $ 1% 1 61 (-o*er ba.. choices say) is a sequence ' may or may not *ish to sum the sequence or

,orm its -rouct 4a, Sarato3a &+

e-.y

o 4ick Fortis says



27/42


28/42

17 0e says

4o2ember 1! !"1% at 11:%1 -m

'N2e a.so notice that the ratio o, successi2e -airs o, numbers in other sum sequences con2er3e

as *e.. For e8am-.e take any three numbers an sum them to make a ,ourth then continuesummin3 the .ast three numbers in the sequence to make the ne8t 0he ratio o, successi2e -airs

o, numbers in this sequence con2er3es on 1%7!$@55!161$9

Simi.ar.y summin3 the .ast ,our ,i2e si8 se2en an ei3ht numbers con2er3e on i,,erent 2a.ues

*hich themse.2es a--ear to con2er3e on !" as you increase the quantity o, numbers *hich are

summe ieA

4umbers &on2er3ent 2a.ue

Summe

! 1$1"%%7@6779

% 1%7!$@55!161$96 17!@5$17@56!7%9

5 17$576!%$$65679

$ 17%5!6%6!6%%9@ 17717$617$$"5"%9

177$"%11@7@%5619

e3ars

0e

e-.y

http://www.goldennumber.net/fibonacci-series/#comment-643http://www.goldennumber.net/fibonacci-series/?replytocom=643#respondhttp://www.goldennumber.net/fibonacci-series/#comment-643http://www.goldennumber.net/fibonacci-series/?replytocom=643#respond


29/42

o 0e says

4o2ember 1 !"1% at :1 -m

' 3uess ' shou. ha2e #oo3.e this ear.ierA

htt-://en*iki-eiaor3/*iki/#enera.iKationsLo,LFibonacciLnumbers

e3ars

0e

e-.y

!" Mahesh chanra says


30/42

!1 Kubaia says


31/42

!! 4 < Smith says

February !5 !"16 at $:!7 -m

Mr Ha*thorneNs comment is interestin3 es-ecia..y *ith res-ect to ictionary e,initions

ne sees that not a.. sequences can be 3enerate by a ,unction

0he ranom sequence is one such (-3 !6@ Mathematics ictionary


32/42

February !5 !"16 at $:%1 -m

Mr Ha*thorneNs comment is interestin3 es-ecia..y *ith res-ect to ictionary e,initions

ne sees that not a.. sequences can be 3enerate by a ,unction

0he ranom sequence is one such (-3 !6@ Mathematics ictionary


33/42


34/42

!$ ee-Cyoti eb says

+u3ust !6 !"16 at 11:1 -m

can u -.s te.. me at *hich 'nian or in *hich 'nian book -hi is isco2ere 1st

e-.y

!@


35/42

o #ary Meisner says

Se-tember ! !"16 at 11:61 -m

ike many other *ors in the Gn3.ish .an3ua3e the ans*er e-ens on *ho you ask an *here you

ask it See htt-://***3o.ennumbernet/-ronouncin3A-hi/ ,or a more in e-th iscussion

e-.y

! Wes.ey Horton says

ctober !1 !"16 at 1!:%1 -m

Tuestion:

', you -ick a ranom number 4 (.ets say 1@) an 4=1 (1) an starte the sequence ,rom those

t*o numbers oes the series con2er3e on -hi or some other in,inite series?

e-.y

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o #ary Meisner says

ctober !1 !"16 at 7:16 -m

You can start *ith any t*o numbers a then to3ether an continue in the same *ay an the ratio o,

the .ar3er to the sma..er *i.. con2er3e on -hi

e-.y

4MQ says

May %" !"1$ at 11:56 am

0he sequence o, e8-onentia. -o*ers o, -hi oes ha2e unique -ro-erties but technica..y s-eakin3 it is no

the sequence isco2ere by Fibonacci an name a,ter him

e-.y

http://www.goldennumber.net/http://www.goldennumber.net/fibonacci-series/#comment-1857http://www.goldennumber.net/fibonacci-series/?replytocom=1857#respondhttp://www.goldennumber.net/fibonacci-series/#comment-3061http://www.goldennumber.net/fibonacci-series/?replytocom=3061#respondhttp://www.goldennumber.net/http://www.goldennumber.net/fibonacci-series/#comment-1857http://www.goldennumber.net/fibonacci-series/?replytocom=1857#respondhttp://www.goldennumber.net/fibonacci-series/#comment-3061http://www.goldennumber.net/fibonacci-series/?replytocom=3061#respond


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!7 ichar +..an QretKschmar ''/


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1818!8%858 0H'0GG4=0WG40Y4G

0he -rouct o, the $ ,irst Fibos 3i2es the sum o, 3ematrias o, the ! ne8t terms

See the 2eri,ication here htt-://***3e,,ree,r/3em-h-?te8te4G=4G=0W=0HGG=F'G=G'#H00H'0GG4=0WG40Y4G

e-.y

o #ary Meisner says

! & % 6 etc 0his *orks ,or the Fibonacci numbers in Gn3.ish What about other .an3ua3es?

e-.y

o '#4E says

http://www.gef.free.fr/gem.php?texte=ONE+ONE+TWO+THREE+FIVE+EIGHTTHIRTEEN+TWENTYONEhttp://www.gef.free.fr/gem.php?texte=ONE+ONE+TWO+THREE+FIVE+EIGHTTHIRTEEN+TWENTYONEhttp://www.goldennumber.net/fibonacci-series/?replytocom=2085#respondhttp://www.goldennumber.net/http://www.goldennumber.net/fibonacci-series/#comment-2087http://www.goldennumber.net/fibonacci-series/?replytocom=2087#respondhttp://www.gef.free.fr/gem.php?texte=ONE+ONE+TWO+THREE+FIVE+EIGHTTHIRTEEN+TWENTYONEhttp://www.gef.free.fr/gem.php?texte=ONE+ONE+TWO+THREE+FIVE+EIGHTTHIRTEEN+TWENTYONEhttp://www.goldennumber.net/fibonacci-series/?replytocom=2085#respondhttp://www.goldennumber.net/http://www.goldennumber.net/fibonacci-series/#comment-2087http://www.goldennumber.net/fibonacci-series/?replytocom=2087#respond


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'Nm no mathematician or scientist but ,rom *hat ' unerstan about braAket notation Cust about

e2erythin3 3ro*s an then ecays accorin3 to .o3arithmic s-ira.s an *hir.in3 squares re-resente

by PS' an PH'

+n *hat 'N2e rea seems to say that there are other -ossib.e .o3arithmic s-ira.s the uni2erse cou. be base

on but the PH' s-ira. is the s.o*est o, a.. an usin3 any o, the others *ou. ha2e mae it im-ossib.e

,or .i,e on earth to e8ist at a..B +ny e8-ert o-inions out there to she more .i3ht on this notion?

e-.y

%! > says

February !6 !"1$ at 1!:!7 am

Hear o, Hy-atia?

e-.y

%eave a 4e&ly

Your emai. aress *i.. not be -ub.ishe equire ,ie.s are marke V



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