EMAT 6680 :: Clay Kitchings :: Assignment 12 :: Fibonacci Numbers

 

 

Exploring Fibonacci Numbers

 

Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1, and f(n) = f(n-1) + f(n-2).

 

* = values begin with n=2

 

n

 

ratios

 

f(n-1) + f(n-2)

* f(n)/f(n-1)

0

1

undefined

1

1

1

2

2

2

3

3

1.5

4

5

1.666666667

5

8

1.6

6

13

1.625

7

21

1.615384615

8

34

1.619047619

9

55

1.617647059

10

89

1.618181818

11

144

1.617977528

12

233

1.618055556

13

377

1.618025751

14

610

1.618037135

15

987

1.618032787

16

1597

1.618034448

17

2584

1.618033813

18

4181

1.618034056

19

6765

1.618033963

20

10946

1.618033999

21

17711

1.618033985

22

28657

1.61803399

23

46368

1.618033988

24

75025

1.618033989

25

121393

1.618033989

26

196418

1.618033989

27

317811

1.618033989

28

514229

1.618033989

29

832040

1.618033989

30

1346269

1.618033989

 

It appears that this sequence converges to approximately 1.61803989...

 

LetŐs see if we can find an exact value for this approximation.  First, letŐs assume that this sequence does indeed converge to some .  Take a ratio for a high value of n as n approaches infinity.

 

The sequence might appear as follows:

 

1, 1, 2, 3, 5, 8, É, a, b, a+b

 

Consider the ratio: 

 

If the sequence converges, it must converge to some value =b/a (b/a is practically equal to (a+b)/b when n approaches + infinity).  This implies:

 

 

 

Now, use the quadratic formula to solve:

 

The Fibonacci Sequence converges to .

 

 

If we take different ratios, we can conjecture that other such ratios converge as well (to different ).

 

n

 

ratios

every 2nd ratio

every 3rd ratio

 

*f(n-1) + f(n-2)

* f(n)/f(n-1)

 

 

0

1

undefined

 

 

1

1

1

 

 

2

2

2

2

 

3

3

1.5

3

3

4

5

1.666666667

2.5

5

5

8

1.6

2.666666667

4

6

13

1.625

2.6

4.333333333

7

21

1.615384615

2.625

4.2

8

34

1.619047619

2.615384615

4.25

9

55

1.617647059

2.619047619

4.230769231

10

89

1.618181818

2.617647059

4.238095238

11

144

1.617977528

2.618181818

4.235294118

12

233

1.618055556

2.617977528

4.236363636

13

377

1.618025751

2.618055556

4.235955056

14

610

1.618037135

2.618025751

4.236111111

15

987

1.618032787

2.618037135

4.236051502

16

1597

1.618034448

2.618032787

4.236074271

17

2584

1.618033813

2.618034448

4.236065574

18

4181

1.618034056

2.618033813

4.236068896

19

6765

1.618033963

2.618034056

4.236067627

20

10946

1.618033999

2.618033963

4.236068111

21

17711

1.618033985

2.618033999

4.236067926

22

28657

1.61803399

2.618033985

4.236067997

23

46368

1.618033988

2.61803399

4.23606797

24

75025

1.618033989

2.618033988

4.23606798

25

121393

1.618033989

2.618033989

4.236067976

26

196418

1.618033989

2.618033989

4.236067978

27

317811

1.618033989

2.618033989

4.236067977

28

514229

1.618033989

2.618033989

4.236067978

29

832040

1.618033989

2.618033989

4.236067977

30

1346269

1.618033989

2.618033989

4.236067978