Clay Bennett

Assignment 12.4

For this assignment, I have chose to investigate the Fibbonnaci sequence.  I had never heard of the Fibbonnaci sequence until my third year of college, so there is a lot I do not know.   At this point all I do know is that it is generated by f(n) = f(n-1) + f(n-2).  I will begin by generating the Fibbonnaci sequence on a spreadsheet.  I will generate it up to f(25) and such that f(0) = 1 and f(1)= 1. 

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

 

Now that we can see how the Fibbonnaci sequence is constructed and what it looks like, we can begin to investigate it.  I will start by taking the ratio of the adjacent terms or more formally g(n) = f(n-1)/f(n-2).

1

 

1

1

2

2

3

1.5

5

1.666666667

8

1.6

13

1.625

21

1.615384615

34

1.619047619

55

1.617647059

89

1.618181818

144

1.617977528

233

1.618055556

377

1.618025751

610

1.618037135

987

1.618032787

1597

1.618034448

2584

1.618033813

4181

1.618034056

6765

1.618033963

10946

1.618033999

17711

1.618033985

28657

1.61803399

46368

1.618033988

75025

1.618033989

 

As we can see, as f(n) gets larger g(n) approaches 1.618, which is the golden ratio.  This was not too surprising to me because I had seen it before, I just did not remember it.  Next, I start playing around with the f(0) and f(1) terms.  I changed them to one and three, as the assignment recommended.  I then took the ratio of its adjacent terms.

1

 

3

3

4

1.333333

7

1.75

11

1.571429

18

1.636364

29

1.611111

47

1.62069

76

1.617021

123

1.618421

199

1.617886

322

1.61809

521

1.618012

843

1.618042

1364

1.618031

2207

1.618035

3571

1.618034

5778

1.618034

9349

1.618034

15127

1.618034

24476

1.618034

39603

1.618034

64079

1.618034

103682

1.618034

167761

1.618034

 

This surprised me somewhat, but not too much because I figured it was a special case.  I then began to plug in various numbers for f(0) and f(1).  To my surprise, the golden ratio came out every time.  Here are a couple of examples…

F(0) = 1 and f(1) = 3

1

 

3

3

4

1.333333

7

1.75

11

1.571429

18

1.636364

29

1.611111

47

1.62069

76

1.617021

123

1.618421

199

1.617886

322

1.61809

521

1.618012

843

1.618042

1364

1.618031

2207

1.618035

3571

1.618034

5778

1.618034

9349

1.618034

15127

1.618034

24476

1.618034

39603

1.618034

64079

1.618034

103682

1.618034

167761

1.618034

 

F(0) = 0 and f(1) = 2

0

 

2

 

2

1

4

2

6

1.5

10

1.666667

16

1.6

26

1.625

42

1.615385

68

1.619048

110

1.617647

178

1.618182

288

1.617978

466

1.618056

754

1.618026

1220

1.618037

1974

1.618033

3194

1.618034

5168

1.618034

8362

1.618034

13530

1.618034

21892

1.618034

35422

1.618034

57314

1.618034

92736

1.618034


F(0) = 300 and f(1) = 1

300

 

1

0.003333

301

301

302

1.003322

603

1.996689

905

1.500829

1508

1.666298

2413

1.600133

3921

1.624948

6334

1.615404

10255

1.61904

16589

1.61765

26844

1.618181

43433

1.617978

70277

1.618055

113710

1.618026

183987

1.618037

297697

1.618033

481684

1.618034

779381

1.618034

1261065

1.618034

2040446

1.618034

3301511

1.618034

5341957

1.618034

8643468

1.618034

 

F(0) = -6 and f(1) = -2

-6

 

-2

0.333333

-8

4

-10

1.25

-18

1.8

-28

1.555556

-46

1.642857

-74

1.608696

-120

1.621622

-194

1.616667

-314

1.618557

-508

1.617834

-822

1.61811

-1330

1.618005

-2152

1.618045

-3482

1.61803

-5634

1.618036

-9116

1.618033

-14750

1.618034

-23866

1.618034

-38616

1.618034

-62482

1.618034

-101098

1.618034

-163580

1.618034

-264678

1.618034

 

Now I will try taking the ratio of every other term or h(n) = f(n-1)/f(n-3).  I have no idea what the ratio will be. 

F(0) = 1 and f(1) = 1

1

 

1

 

2

2

3

3

5

2.5

8

2.666667

13

2.6

21

2.625

34

2.615385

55

2.619048

89

2.617647

144

2.618182

233

2.617978

377

2.618056

610

2.618026

987

2.618037

1597

2.618033

2584

2.618034

4181

2.618034

6765

2.618034

10946

2.618034

17711

2.618034

28657

2.618034

46368

2.618034

75025

2.618034

 

As you can see, the ratio seems to be the golden ratio plus one.  Now I am going to check to see if this works for other values of f(0) and f(1).

F(0) = 0 and f(1) = 2

0

 

2

 

2

#DIV/0!

4

2

6

3

10

2.5

16

2.666667

26

2.6

42

2.625

68

2.615385

110

2.619048

178

2.617647

288

2.618182

466

2.617978

754

2.618056

1220

2.618026

1974

2.618037

3194

2.618033

5168

2.618034

8362

2.618034

13530

2.618034

21892

2.618034

35422

2.618034

57314

2.618034

92736

2.618034

 

F(0) = -7 and f(1) = -99

-7

 

-99

 

-106

15.14286

-205

2.070707

-311

2.933962

-516

2.517073

-827

2.659164

-1343

2.602713

-2170

2.623942

-3513

2.615786

-5683

2.618894

-9196

2.617706

-14879

2.618159

-24075

2.617986

-38954

2.618052

-63029

2.618027

-101983

2.618037

-165012

2.618033

-266995

2.618034

-432007

2.618034

-699002

2.618034

-1131009

2.618034

-1830011

2.618034

-2961020

2.618034

-4791031

2.618034

 

F(0) = .001 and f(1) = .35

0.35

 

0.001

 

0.351

1.002857

0.352

352

0.703

2.002849

1.055

2.997159

1.758

2.500711

2.813

2.666351

4.571

2.600114

7.384

2.624956

11.955

2.615401

19.339

2.619041

31.294

2.61765

50.633

2.618181

81.927

2.617978

132.56

2.618055

214.487

2.618026

347.047

2.618037

561.534

2.618033

908.581

2.618034

1470.115

2.618034

2378.696

2.618034

3848.811

2.618034

6227.507

2.618034

10076.32

2.618034

 

F(0) = -654 and f(1) = 1.2

-654

 

1.2

 

-652.8

0.998165

-651.6

-543

-1304.4

1.998162

-1956

3.001842

-3260.4

2.49954

-5216.4

2.666871

-8476.8

2.599926

-13693.2

2.625029

-22170

2.615374

-35863.2

2.619052

-58033.2

2.617645

-93896.4

2.618182

-151930

2.617977

-245826

2.618056

-397756

2.618026

-643582

2.618037

-1041337

2.618033

-1684919

2.618034

-2726256

2.618034

-4411175

2.618034

-7137431

2.618034

-1.2E+07

2.618034

-1.9E+07

2.618034

 

I tried to pick the most random values for f(0) and f(1) and no matter what their value was the ratio was always the gold ratio plus one. 

My final conclusions for this assignment are…

F(n) = f(n-1) + f(n-2)  => f(n-1)/f(n-2) = 1.61… = (1 + sqrt(5)) / 2 = the golden ratio.

F(n) = f(n-1) + f(n-2)  => f(n-1)/f(n-3) = 2.61… =[(1 + sqrt(5)) / 2] + 1 = the golden ratio + 1.