Matematik arbetsblad förskolan

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The spiral begins from her left wrist and travels to the background of the painting that follows the sequence.

Solved Examples

Find the sum of the first 15 Fibonacci numbers.

As we know,
The sum of the Fibonacci Sequence = ${\sum ^{n}_{i=0}F_{i} = F_{n+2} – F_{2}}$
= ${F_{n+2}-1}$, where Fn is the nth Fibonacci number, and the sequence starts from F0.
Thus, the sum of the first 15 Fibonacci numbers = (15+2)th term – 2nd term
= 17th term – 1
= 987 – 1
= 986

Find the 5th Fibonacci number.

As we know,
The nth Fibonacci number is F(xn ) = F(xn-1) + F(xn-2), for n>2
Then the 5th Fibonacci number is F(x5 ) = F(x5-1) + F(x5-2), for n=5
= F(x4) + F(x3)
= 2 + 1
= 3

Find the next number when F14 = 377.

Here,
F15 = F14 x Golden ratio
= 377 x 1.618034 (up to 4 decimals)
= 609.9988 (up to 4 decimals), which is approximately 610
Hence, F15 = 610

Calculate the value of F-6.

As we know,
F-n = (-1)n+1 Fn
Here,
F-6 = (-1)6+1 F6
= (-1) x 5
= -5

Last modified on June 10th, 2024

Fibonacci Sequence Visualisor

Designed for Casey Mongoven as a tool for his music composition, this page will calculate large numbers of terms of various sequencesthat are related to the Fibonacci Numbers and display them in agraph-like form so that they are easily visualised.

:

01123581321345589144233377610...

Why?

Because adding two odd numbers produces an even number, but adding even and odd (in any order) produces an odd number.


Lucas Numbers

Starting the sequence with 2 and 1 we get the "Lucas Numbers".

Let us try a few:

a

b

 

b / a

2

3

 1.5

3

5

 1.666666666...

5

8

 1.6

8

13

 1.625

...

...

 ...

144

233

 1.618055556...

233

377

 1.618025751...

...

...

 ...

Try some start values yourself:

images/fibo.js

It can take longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

xn=φn − (1−φ)n√5

The answer comes out as a whole number, exactly equal to the addition of the previous two terms.

Example: x6

x6=(1.618034...)6 − (1−1.618034...)6√5

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033 , a more accurate calculation would be closer to 8.

Try n=12 and see what you get.

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

Example: What is the next in the sequence after 8 ?

It will be 8 times φ:

8φ = 8 × 1.618034...
= 12.94427...
=13 (rounded)

Some Interesting Things

An odd fact:

The sequence goes even, odd, odd, even, odd, odd, even, odd, odd, ...

No value is repeated repeated in the sums. Each real number has a unique spectrum. #reps means the number of representations. Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. Try it yourself!

Here is the Fibonacci sequence again:

n =0123456789101112131415...
xn=01123581321345589144233377610...

There is an interesting pattern:

  • Look at the number x3= 2.

    If x is irrational then no exact values will ever be repeated in the table. Every 4th number is a multiple of 3 (3, 21,144, ...)

  • Look at the number x5= 5. The series numbers are shown in the Sequence Valuesbox and in the Sequence Plot they are translated into squares (give the size in pixels) on a "graph" where the direction ofthe graph's base line is selected (>=across, v=down, ^=up) and the numbers becomes distances from that base line.

    The Spectrum and Signature numbers can be expressions such as sqrt(2) or sin(Pi/3) (trig functions use radian meaure).

    Phi = 1.6180339...
    We now sort the entries in the table intoincreasing order. The sequence formed by Fibonacci numbers is called the Fibonacci sequence.

    The following is a full list of the first 10, 100, and 300 Fibonacci numbers.

    The First 10 Fibonacci Numbers

    1. 1

    2. 1

    3. 2

    4. 3

    5. 5

    6. 8

    7. 13

    8. 21

    9. 34

    10. 55

    The First 100 Fibonacci Numbers

    The first 100 Fibonacci numbers includes the Fibonacci numbers above and the numbers in this section.

    11. 89

    12. 144

    13. 233

    14. 377

    15. 610

    16. 987

    17. 1597

    18. 2584

    19. 4181

    20. 6765

    21. 10946

    22. 17711

    23. 28657

    24. 46368

    25. 75025

    26. 121393

    27. 196418

    28. 317811

    29. 514229

    30. 832040

    31. 1346269

    32. 2178309

    33. 3524578

    34. 5702887

    35. 9227465

    36. 14930352

    37. 24157817

    38. 39088169

    39. 63245986

    40. 102334155

    41. 165580141

    42. 267914296

    43. 433494437

    44. 701408733

    45. 1134903170

    46. 1836311903

    47. 2971215073

    48. 4807526976

    49. 7778742049

    50. 12586269025

    51. 20365011074

    52. 32951280099

    53. 53316291173

    54. 86267571272

    55. 139583862445

    56. 225851433717

    57. 365435296162

    58. 591286729879

    59. 956722026041

    60. 1548008755920

    61. 2504730781961

    62. 4052739537881

    63. 6557470319842

    64. 10610209857723

    65. 17167680177565

    66. 27777890035288

    67. 44945570212853

    68. 72723460248141

    69. 117669030460994

    70. 190392490709135

    71. 308061521170129

    72. 498454011879264

    73. 806515533049393

    74. 1304969544928657

    75. 2111485077978050

    76. 3416454622906707

    77. 5527939700884757

    78. 8944394323791464

    79. 14472334024676221

    80. 23416728348467685

    81. 37889062373143906

    82. 61305790721611591

    83. 99194853094755497

    84. 160500643816367088

    85. 259695496911122585

    86. 420196140727489673

    87. 679891637638612258

    88. 1100087778366101931

    89. 1779979416004714189

    90. 2880067194370816120

    91. 4660046610375530309

    92. 7540113804746346429

    93. 12200160415121876738

    94. 19740274219868223167

    95. 31940434634990099905

    96. 51680708854858323072

    97. 83621143489848422977

    98. 135301852344706746049

    99. 218922995834555169026

    100. 354224848179261915075

    The First 300 Fibonacci Numbers

    The first 300 Fibonacci numbers includes the Fibonacci numbers above and the numbers below.

    101. 573147844013817084101

    102. 927372692193078999176

    103. 1500520536206896083277

    104. 2427893228399975082453

    105. 3928413764606871165730

    106. 6356306993006846248183

    107. 10284720757613717413913

    108. 16641027750620563662096

    109. 26925748508234281076009

    110. 43566776258854844738105

    111. 70492524767089125814114

    112. 114059301025943970552219

    113. 184551825793033096366333

    114. 298611126818977066918552

    115. 483162952612010163284885

    116. 781774079430987230203437

    117. 1264937032042997393488322

    118. 2046711111473984623691759

    119. 3311648143516982017180081

    120. 5358359254990966640871840

    121. 8670007398507948658051921

    122. 14028366653498915298923761

    123. 22698374052006863956975682

    124. 36726740705505779255899443

    125. 59425114757512643212875125

    126. 96151855463018422468774568

    127. 155576970220531065681649693

    128. 251728825683549488150424261

    129. 407305795904080553832073954

    130. 659034621587630041982498215

    131. 1066340417491710595814572169

    132. 1725375039079340637797070384

    133. 2791715456571051233611642553

    134. 4517090495650391871408712937

    135. 7308805952221443105020355490

    136. 11825896447871834976429068427

    137. 19134702400093278081449423917

    138. 30960598847965113057878492344

    139. 50095301248058391139327916261

    140. 81055900096023504197206408605

    141. 131151201344081895336534324866

    142. 212207101440105399533740733471

    143. 343358302784187294870275058337

    144. 555565404224292694404015791808

    145. 898923707008479989274290850145

    146. 1454489111232772683678306641953

    147. 2353412818241252672952597492098

    148. 3807901929474025356630904134051

    149. 6161314747715278029583501626149

    150. 9969216677189303386214405760200

    151. 16130531424904581415797907386349

    152. 26099748102093884802012313146549

    153. 42230279526998466217810220532898

    154. 68330027629092351019822533679447

    155. 110560307156090817237632754212345

    156. 178890334785183168257455287891792

    157. 289450641941273985495088042104137

    158. 468340976726457153752543329995929

    159. 757791618667731139247631372100066

    160. 1226132595394188293000174702095995

    161. 1983924214061919432247806074196061

    162. 3210056809456107725247980776292056

    163. 5193981023518027157495786850488117

    164. 8404037832974134882743767626780173

    165. 13598018856492162040239554477268290

    166. 22002056689466296922983322104048463

    167. 35600075545958458963222876581316753

    168. 57602132235424755886206198685365216

    169. 93202207781383214849429075266681969

    170. 150804340016807970735635273952047185

    171. 244006547798191185585064349218729154

    172. 394810887814999156320699623170776339

    173. 638817435613190341905763972389505493

    174. 1033628323428189498226463595560281832

    175. 1672445759041379840132227567949787325

    176. 2706074082469569338358691163510069157

    177. 4378519841510949178490918731459856482

    178. 7084593923980518516849609894969925639

    179. 11463113765491467695340528626429782121

    180. 18547707689471986212190138521399707760

    181. 30010821454963453907530667147829489881

    182. 48558529144435440119720805669229197641

    183. 78569350599398894027251472817058687522

    184. 127127879743834334146972278486287885163

    185. 205697230343233228174223751303346572685

    186. 332825110087067562321196029789634457848

    187. 538522340430300790495419781092981030533

    188. 871347450517368352816615810882615488381

    189. 1409869790947669143312035591975596518914

    190. 2281217241465037496128651402858212007295

    191. 3691087032412706639440686994833808526209

    192. 5972304273877744135569338397692020533504

    193. 9663391306290450775010025392525829059713

    194. 15635695580168194910579363790217849593217

    195. 25299086886458645685589389182743678652930

    196. 40934782466626840596168752972961528246147

    197. 66233869353085486281758142155705206899077

    198. 107168651819712326877926895128666735145224

    199. 173402521172797813159685037284371942044301

    200. 280571172992510140037611932413038677189525

    201. 453973694165307953197296969697410619233826

    202. 734544867157818093234908902110449296423351

    203. 1188518561323126046432205871807859915657177

    204. 1923063428480944139667114773918309212080528

    205. 3111581989804070186099320645726169127737705

    206. 5034645418285014325766435419644478339818233

    207. 8146227408089084511865756065370647467555938

    208. 13180872826374098837632191485015125807374171

    209. 21327100234463183349497947550385773274930109

    210. 34507973060837282187130139035400899082304280

    211. 55835073295300465536628086585786672357234389

    212. 90343046356137747723758225621187571439538669

    213. 146178119651438213260386312206974243796773058

    214. 236521166007575960984144537828161815236311727

    215. 382699285659014174244530850035136059033084785

    216. 619220451666590135228675387863297874269396512

    217. 1001919737325604309473206237898433933302481297

    218. 1621140188992194444701881625761731807571877809

    219. 2623059926317798754175087863660165740874359106

    220. 4244200115309993198876969489421897548446236915

    221. 6867260041627791953052057353082063289320596021

    222. 11111460156937785151929026842503960837766832936

    223. 17978720198565577104981084195586024127087428957

    224. 29090180355503362256910111038089984964854261893

    225. 47068900554068939361891195233676009091941690850

    226. 76159080909572301618801306271765994056795952743

    227. 123227981463641240980692501505442003148737643593

    228. 199387062373213542599493807777207997205533596336

    229. 322615043836854783580186309282650000354271239929

    230. 522002106210068326179680117059857997559804836265

    231. 844617150046923109759866426342507997914076076194

    232. 1366619256256991435939546543402365995473880912459

    233. 2211236406303914545699412969744873993387956988653

    234. 3577855662560905981638959513147239988861837901112

    235. 5789092068864820527338372482892113982249794889765

    236. 9366947731425726508977331996039353971111632790877

    237. 15156039800290547036315704478931467953361427680642

    238. 24522987531716273545293036474970821924473060471519

    239. 39679027332006820581608740953902289877834488152161

    240. 64202014863723094126901777428873111802307548623680

    241. 103881042195729914708510518382775401680142036775841

    242. 168083057059453008835412295811648513482449585399521

    243. 271964099255182923543922814194423915162591622175362

    244. 440047156314635932379335110006072428645041207574883

    245. 712011255569818855923257924200496343807632829750245

    246. 1152058411884454788302593034206568772452674037325128

    247. 1864069667454273644225850958407065116260306867075373

    248. 3016128079338728432528443992613633888712980904400501

    249. 4880197746793002076754294951020699004973287771475874

    250. 7896325826131730509282738943634332893686268675876375

    251. 12776523572924732586037033894655031898659556447352249

    252. 20672849399056463095319772838289364792345825123228624

    253. 33449372971981195681356806732944396691005381570580873

    254. 54122222371037658776676579571233761483351206693809497

    255. 87571595343018854458033386304178158174356588264390370

    256. 141693817714056513234709965875411919657707794958199867

    257. 229265413057075367692743352179590077832064383222590237

    258. 370959230771131880927453318055001997489772178180790104

    259. 600224643828207248620196670234592075321836561403380341

    260. 971183874599339129547649988289594072811608739584170445

    261. 1571408518427546378167846658524186148133445300987550786

    262. 2542592393026885507715496646813780220945054040571721231

    263. 4114000911454431885883343305337966369078499341559272017

    264. 6656593304481317393598839952151746590023553382130993248

    265. 10770594215935749279482183257489712959102052723690265265

    266. 17427187520417066673081023209641459549125606105821258513

    267. 28197781736352815952563206467131172508227658829511523778

    268. 45624969256769882625644229676772632057353264935332782291

    269. 73822750993122698578207436143903804565580923764844306069

    270. 119447720249892581203851665820676436622934188700177088360

    271. 193270471243015279782059101964580241188515112465021394429

    272. 312718191492907860985910767785256677811449301165198482789

    273. 505988662735923140767969869749836918999964413630219877218

    274. 818706854228831001753880637535093596811413714795418360007

    275. 1324695516964754142521850507284930515811378128425638237225

    276. 2143402371193585144275731144820024112622791843221056597232

    277. 3468097888158339286797581652104954628434169971646694834457

    278. 5611500259351924431073312796924978741056961814867751431689

    279. 9079598147510263717870894449029933369491131786514446266146

    280. 14691098406862188148944207245954912110548093601382197697835

    281. 23770696554372451866815101694984845480039225387896643963981

    282. 38461794961234640015759308940939757590587318989278841661816

    283. 62232491515607091882574410635924603070626544377175485625797

    284. 100694286476841731898333719576864360661213863366454327287613

    285. 162926777992448823780908130212788963731840407743629812913410

    286. 263621064469290555679241849789653324393054271110084140201023

    287. 426547842461739379460149980002442288124894678853713953114433

    288. 690168906931029935139391829792095612517948949963798093315456

    289. 1116716749392769314599541809794537900642843628817512046429889

    290. 1806885656323799249738933639586633513160792578781310139745345

    291. 2923602405716568564338475449381171413803636207598822186175234

    292. 4730488062040367814077409088967804926964428786380132325920579

    293. 7654090467756936378415884538348976340768064993978954512095813

    294. 12384578529797304192493293627316781267732493780359086838016392

    295. 20038668997554240570909178165665757608500558774338041350112205

    296. 32423247527351544763402471792982538876233052554697128188128597

    297. 52461916524905785334311649958648296484733611329035169538240802

    298. 84885164052257330097714121751630835360966663883732297726369399

    299. 137347080577163115432025771710279131845700275212767467264610201

    300. 222232244629420445529739893461909967206666939096499764990979600

The pattern of seeds in the sunflower also follows this sequence.
  • We can find this sequence in rabbits.

    8 is 8 itself or 3+5 or 1+2+5. They get closer and closer to the powers (exponents) of the Golden Ratio:

    Lucas
    Number    		nφn
    201.0000...
    111.6180...
    322.6180...
    434.2361...
    746.8541...
    11511.0902...
    18617.9443...
    29729.0344...
    47846.9787...
    76976.0132...
    12310122.9919...
    19911199.0050...
    .........

    For example, the 15th Lucas Number is approximately φ15= 1364.0007..., so is exactly 1364.

    and so its spectrum is

    1, 3, 4, 6, 8, ... The row values (c) of each entry in this ordered list is called the signature of x.
    Sorting, with (c,d) being the row and column numbers, we have2.62 at (1,1), 3.62 at (2,1), 4.24 at (1,2), 4.62 at (3,1), 5.24 at (2,2) and so on.The sequence of row numbers (c) in this list is the signature of Phi: 1,2,1,3,2,...
    More at Mathworld .
    © Dr Ron Knott for Casey Mongoven: version 7 February 2005

    • Fibonacci Sequence

    The Fibonacci Sequence is the series of numbers:

    0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

    The next number is found by adding up the two numbers before it:

    • the 2 is found by adding the two numbers before it (1+1),
    • the 3 is found by adding the two numbers before it (1+2),
    • the 5 is (2+3),
    • and so on!

    Example: the next number in the sequence above is 21+34 =55

    It is that simple!

    Here is a longer list:

    0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, ...

    Can you figure out the next few numbers?

    Makes A Spiral

    We van make squares with fibonacci side lengths, place them next to each other, and we get a nice spiral:


    The Fibonacci Spiral
    See how the squares fit neatly together?
    For example 5 and 8 make 13, 8 and 13 make 21, and so on.


    This spiral is found in nature!
    See: Nature, The Golden Ratio, and Fibonacci

    The Rule

    The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series).

    First, the terms are numbered from 0 onwards like this:

    n =xn=

    00

    11

    21

    32

    43

    55

    68

    713

    821

    934

    1055

    1189

    12144

    13233

    14377

    ...

    So term number 6 is called x6 (which equals 8).

    Example: the 8th term is
    the 7th term plus the 6th term:


    x8= x7 + x6

    So we can write the rule:

    xn= xn−1 + xn−2

    where:

    • xn is term number "n"
    • xn−1 is the previous term (n−1)
    • xn−2 is the term before that (n−2)

    Example: term 9 is calculated like this:

    x9= x9−1 + x9−2= x8 + x7= 21 + 13 = 34

    Rules like this are called recursive formulas.

    To use a recursive formula we also need to know the first few terms.

    It can be written like this:

    x−n= (−1)n+1xn

    Which says term "−n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, ... The numbers in this sequence, known as the Fibonacci numbers, are denoted by Fn.

    The first few numbers of the Fibonacci Sequence are as follows.

    Formula

    The above sequence can be written as a ‘Rule’, which is expressed with the following equation.

    Using this equation, we can conclude that the sequence continues to infinity.

    The following table lists each term and term value in the Fibonacci Sequence till the 10th.

    Term PositionFn Term Term Value / Fibonacci number
    1stF00
    2ndF11
    3rdF21
    4thF32
    5thF43
    6thF55
    7thF68
    8thF713
    9thF821
    10thF934

    Patterns

    The numbers in the sequence follow some interesting patterns: 

    • Every third number in the series, starting at 2, is a multiple of 2.

      matematik arbetsblad förskolan

      Phi means (sqrt(5)+1)/2 and phi is (sqrt(5)-1)/2.

      Click on a Sequence name button to see the Sequence Plot and Values.

      Notes on the sequences themselves and what they meanare lower down on this page.

      Sequence Plot:
      Vertical Para-Fib
      A(n) is the row number of n in the Wythoff array
      Horizontal Para-Fib
      A(n) is the column number of n in the Wythoff array
      R(n)
      is the number of sets of Fibonacci numbers (1,2,3,5,8,13...) that sum to n, e.g.

      Here, the middle numbers of each row are the sum of the two numbers above it.

      Fibonacci Sequence in Real-Life

      In Nature

      We find the Fibonacci Sequence in various fields, from nature to the human body.  

      • It appears in plants with many seed heads, pinecones, fruits, and vegetables.

        E.g. if x is Phi=1.618..

        Fibonacci Sequence

        The Fibonacci Sequence is a number series in which each number is obtained by adding its two preceding numbers. pattern.