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Assignment 12 :: Spreadsheets

 

Fibonacci Sequence

 

By Jamie K. York

 

fibonacci.png

 


 

Have you ever seen a pattern in nature? Perhaps you have seen one in a seashell or a cactus. Many of these patterns can be described mathematically using sequences.

 

The Fibonacci sequence is a series of numbers such that f(0) = 1, f(1) = 1, and f(n) = f(n-1) + f(n-2).  See the full list of numbers in the series below, up to f(30):

 

n

f(n)

0

1

1

1

2

2

3

3

4

5

5

8

6

13

7

21

8

34

9

55

10

89

11

144

12

233

13

377

14

610

15

987

16

1597

17

2584

18

4181

19

6765

20

10946

21

17711

22

28657

23

46368

24

75025

25

121393

26

196418

27

317811

28

514229

29

832040

30

1346269

 

In the sequence, there are multiple patterns that exist. LetŐs explore these together.

 

 

Adjacent Terms

 

In a comparison of adjacent terms in the sequence, we can see that the ratio approaches a specific value as n increases.  

 

n

f(n)

Ratio of Adjacent Terms

0

1

1

1

1

0.50000000

2

2

0.66666667

3

3

0.60000000

4

5

0.62500000

5

8

0.61538462

6

13

0.61904762

7

21

0.61764706

8

34

0.61818182

9

55

0.61797753

10

89

0.61805556

11

144

0.61802575

12

233

0.61803714

13

377

0.61803279

14

610

0.61803445

15

987

0.61803381

16

1597

0.61803406

17

2584

0.61803396

18

4181

0.61803400

19

6765

0.61803399

20

10946

0.61803399

21

17711

0.61803399

22

28657

0.61803399

23

46368

0.61803399

24

75025

0.61803399

25

121393

0.61803399

26

196418

0.61803399

27

317811

0.61803399

28

514229

0.61803399

29

832040

0.61803399

30

1346269

 

 

 

Every Second Term

 

Next we can look at the ratio of every second term. Similarly to the ratio of adjacent terms, we see that this ratio approaches a new value as n increases.

 

n

f(n)

Ratio of Every Second Term

0

1

0.5

1

1

0.333333333

2

2

0.4

3

3

0.375

4

5

0.384615385

5

8

0.380952381

6

13

0.382352941

7

21

0.381818182

8

34

0.382022472

9

55

0.381944444

10

89

0.381974249

11

144

0.381962865

12

233

0.381967213

13

377

0.381965552

14

610

0.381966187

15

987

0.381965944

16

1597

0.381966037

17

2584

0.381966001

18

4181

0.381966015

19

6765

0.38196601

20

10946

0.381966012

21

17711

0.381966011

22

28657

0.381966011

23

46368

0.381966011

24

75025

0.381966011

25

121393

0.381966011

26

196418

0.381966011

27

317811

0.381966011

28

514229

0.381966011

29

832040

 

30

1346269

 

 

 

Sequence Modifications

 

The Fibonacci sequence above is based on the fact that f(0) = 1 and f(1) = 1. What would happen to the pattern if these constants were altered? LetŐs consider other integers greater than 1.

 

If f(0) = 1 and f(1) = 3, this reflects a Lucas Sequence. However, with further analysis of the same two ratios, you will find that they approach the same values. The ratio of adjacent terms approaches 0.61803399 as n increases, and the ratio of every second term approaches 0.381966011.

 

n

f(n)

Ratio of Adjacent Terms

Ratio of Every Second Term

0

1

0.333333333

0.25

1

3

0.75000000

0.428571429

2

4

0.57142857

0.363636364

3

7

0.63636364

0.388888889

4

11

0.61111111

0.379310345

5

18

0.62068966

0.382978723

6

29

0.61702128

0.381578947

7

47

0.61842105

0.382113821

8

76

0.61788618

0.381909548

9

123

0.61809045

0.381987578

10

199

0.61801242

0.381957774

11

322

0.61804223

0.381969158

12

521

0.61803084

0.381964809

13

843

0.61803519

0.38196647

14

1364

0.61803353

0.381965836

15

2207

0.61803416

0.381966078

16

3571

0.61803392

0.381965986

17

5778

0.61803401

0.381966021

18

9349

0.61803398

0.381966008

19

15127

0.61803399

0.381966013

20

24476

0.61803399

0.381966011

21

39603

0.61803399

0.381966011

22

64079

0.61803399

0.381966011

23

103682

0.61803399

0.381966011

24

167761

0.61803399

0.381966011

25

271443

0.61803399

0.381966011

26

439204

0.61803399

0.381966011

27

710647

0.61803399

0.381966011

28

1149851

0.61803399

0.381966011

29

1860498

0.61803399

 

30

3010349

 

 

 

In further investigation, we can explore other modifications to the value of f(1) to find that these two ratios still remain unchanged.

 

f(1) = 4

 

n

f(n)

Ratio of Adjacent Terms

Ratio of Every Second Term

0

1

0.25

0.2

1

4

0.80000000

0.444444444

2

5

0.55555556

0.357142857

3

9

0.64285714

0.391304348

4

14

0.60869565

0.378378378

5

23

0.62162162

0.383333333

6

37

0.61666667

0.381443299

7

60

0.61855670

0.382165605

8

97

0.61783439

0.381889764

9

157

0.61811024

0.381995134

10

254

0.61800487

0.381954887

11

411

0.61804511

0.38197026

12

665

0.61802974

0.381964388

13

1076

0.61803561

0.381966631

14

1741

0.61803337

0.381965774

15

2817

0.61803423

0.381966102

16

4558

0.61803390

0.381965977

17

7375

0.61803402

0.381966024

18

11933

0.61803398

0.381966006

19

19308

0.61803399

0.381966013

20

31241

0.61803399

0.381966011

21

50549

0.61803399

0.381966012

22

81790

0.61803399

0.381966011

23

132339

0.61803399

0.381966011

24

214129

0.61803399

0.381966011

25

346468

0.61803399

0.381966011

26

560597

0.61803399

0.381966011

27

907065

0.61803399

0.381966011

28

1467662

0.61803399

0.381966011

29

2374727

0.61803399

 

30

3842389

 

 

 

f(1) = 10

 

n

f(n)

Ratio of Adjacent Terms

Ratio of Every Second Term

0

1

0.1

0.090909091

1

10

0.90909091

0.476190476

2

11

0.52380952

0.34375

3

21

0.65625000

0.396226415

4

32

0.60377358

0.376470588

5

53

0.62352941

0.384057971

6

85

0.61594203

0.381165919

7

138

0.61883408

0.382271468

8

223

0.61772853

0.381849315

9

361

0.61815068

0.382010582

10

584

0.61798942

0.381948986

11

945

0.61805101

0.381972514

12

1529

0.61802749

0.381963527

13

2474

0.61803647

0.38196696

14

4003

0.61803304

0.381965649

15

6477

0.61803435

0.38196615

16

10480

0.61803385

0.381965958

17

16957

0.61803404

0.381966031

18

27437

0.61803397

0.381966004

19

44394

0.61803400

0.381966014

20

71831

0.61803399

0.38196601

21

116225

0.61803399

0.381966012

22

188056

0.61803399

0.381966011

23

304281

0.61803399

0.381966011

24

492337

0.61803399

0.381966011

25

796618

0.61803399

0.381966011

26

1288955

0.61803399

0.381966011

27

2085573

0.61803399

0.381966011

28

3374528

0.61803399

0.381966011

29

5460101

0.61803399

 

30

8834629

 

 

 

We can also explore modifications to the value of f(0) to find that these two ratios still remain unchanged.

 

f(0) = 5

 

n

f(n)

Ratio of Adjacent Terms

Ratio of Every Second Term

0

5

5

0.833333333

1

1

0.16666667

0.142857143

2

6

0.85714286

0.461538462

3

7

0.53846154

0.35

4

13

0.65000000

0.393939394

5

20

0.60606061

0.377358491

6

33

0.62264151

0.38372093

7

53

0.61627907

0.381294964

8

86

0.61870504

0.382222222

9

139

0.61777778

0.381868132

10

225

0.61813187

0.382003396

11

364

0.61799660

0.381951731

12

589

0.61804827

0.381971466

13

953

0.61802853

0.381963928

14

1542

0.61803607

0.381966807

15

2495

0.61803319

0.381965707

16

4037

0.61803429

0.381966127

17

6532

0.61803387

0.381965967

18

10569

0.61803403

0.381966028

19

17101

0.61803397

0.381966005

20

27670

0.61803400

0.381966014

21

44771

0.61803399

0.38196601

22

72441

0.61803399

0.381966012

23

117212

0.61803399

0.381966011

24

189653

0.61803399

0.381966011

25

306865

0.61803399

0.381966011

26

496518

0.61803399

0.381966011

27

803383

0.61803399

0.381966011

28

1299901

0.61803399

0.381966011

29

2103284

0.61803399

 

30

3403185

 

 

 

 

f(0) = 10

 

n

f(n)

Ratio of Adjacent Terms

Ratio of Every Second Term

0

10

10

0.909090909

1

1

0.09090909

0.083333333

2

11

0.91666667

0.47826087

3

12

0.52173913

0.342857143

4

23

0.65714286

0.396551724

5

35

0.60344828

0.376344086

6

58

0.62365591

0.38410596

7

93

0.61589404

0.381147541

8

151

0.61885246

0.382278481

9

244

0.61772152

0.381846635

10

395

0.61815336

0.382011605

11

639

0.61798839

0.381948595

12

1034

0.61805140

0.381972663

13

1673

0.61802734

0.38196347

14

2707

0.61803653

0.381966982

15

4380

0.61803302

0.381965641

16

7087

0.61803436

0.381966153

17

11467

0.61803385

0.381965957

18

18554

0.61803404

0.381966032

19

30021

0.61803397

0.381966003

20

48575

0.61803400

0.381966014

21

78596

0.61803399

0.38196601

22

127171

0.61803399

0.381966012

23

205767

0.61803399

0.381966011

24

332938

0.61803399

0.381966011

25

538705

0.61803399

0.381966011

26

871643

0.61803399

0.381966011

27

1410348

0.61803399

0.381966011

28

2281991

0.61803399

0.381966011

29

3692339

0.61803399

 

30

5974330

 

 

 

 

Now, we can explore the modification of both variables to see that the ratios still remain constant.

 

f(0) = 5, f(1) = 5

 

n

f(n)

Ratio of Adjacent Terms

Ratio of Every Second Term

0

5

1

0.5

1

5

0.50000000

0.333333333

2

10

0.66666667

0.4

3

15

0.60000000

0.375

4

25

0.62500000

0.384615385

5

40

0.61538462

0.380952381

6

65

0.61904762

0.382352941

7

105

0.61764706

0.381818182

8

170

0.61818182

0.382022472

9

275

0.61797753

0.381944444

10

445

0.61805556

0.381974249

11

720

0.61802575

0.381962865

12

1165

0.61803714

0.381967213

13

1885

0.61803279

0.381965552

14

3050

0.61803445

0.381966187

15

4935

0.61803381

0.381965944

16

7985

0.61803406

0.381966037

17

12920

0.61803396

0.381966001

18

20905

0.61803400

0.381966015

19

33825

0.61803399

0.38196601

20

54730

0.61803399

0.381966012

21

88555

0.61803399

0.381966011

22

143285

0.61803399

0.381966011

23

231840

0.61803399

0.381966011

24

375125

0.61803399

0.381966011

25

606965

0.61803399

0.381966011

26

982090

0.61803399

0.381966011

27

1589055

0.61803399

0.381966011

28

2571145

0.61803399

0.381966011

29

4160200

0.61803399

 

30

6731345

 

 

 

 

Further Exploration

 

How would the pattern change if f(0) and f(1) were changed to non-integer values?

 

How would the pattern change if f(0) and f(1) were changed to irrational values?

 

 


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