An Exploration of the Fibonacci Sequence

By

Kenneth E. Montgomery

Fibonacci sequences are sequences of whole numbers generated by Equation 1:

Equation 1:      , for all

The Fibonacci Sequence is generated by Equation 1, where. A spreadsheet was used to investigate this sequence and the ratios of its terms (Table 1).

n

f(n)

f(n+1)/f(n)

f(n+2)/f(n)

f(n+3)/f(n)

0

1

 

 

 

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

Table 1: The Fibonacci sequence and ratios of its terms

Open the XLS file to explore this problem. In Table 1, we have the first thirty Fibonacci terms and the ratios of sequential terms, every second term and every third term. The values to which these ratios tend to converge are explored.

Careful observation leads to the hypothesis that the ratio of sequential Fibonacci terms approaches the value known as the Golden Ratio and given by:

Experimentation in the spreadsheet file also leads to the realization that:

Which leads us to hypothesize that the ratio of every second term approaches the value:

Again, experimentation within the spreadsheet leads to another realization that:

Thus, we have the ratio of every third term approaching this value, which should be given by:

Setting  and , we have the Lucas Sequence (Table 2).

n

f(n)

f(n+1)/f(n)

f(n+2)/f(n)

f(n+3)/f(n)

0

1

 

 

 

1

3

3

 

 

2

4

1.333333

4

 

3

7

1.75

2.333333

7

4

11

1.571429

2.75

3.666667

5

18

1.636364

2.571429

4.5

6

29

1.611111

2.636364

4.142857

7

47

1.62069

2.611111

4.272727

8

76

1.617021

2.62069

4.222222

9

123

1.618421

2.617021

4.241379

10

199

1.617886

2.618421

4.234043

11

322

1.61809

2.617886

4.236842

12

521

1.618012

2.61809

4.235772

13

843

1.618042

2.618012

4.236181

14

1364

1.618031

2.618042

4.236025

15

2207

1.618035

2.618031

4.236084

16

3571

1.618034

2.618035

4.236062

17

5778

1.618034

2.618034

4.23607

18

9349

1.618034

2.618034

4.236067

19

15127

1.618034

2.618034

4.236068

20

24476

1.618034

2.618034

4.236068

21

39603

1.618034

2.618034

4.236068

22

64079

1.618034

2.618034

4.236068

23

103682

1.618034

2.618034

4.236068

24

167761

1.618034

2.618034

4.236068

25

271443

1.618034

2.618034

4.236068

26

439204

1.618034

2.618034

4.236068

27

710647

1.618034

2.618034

4.236068

28

1149851

1.618034

2.618034

4.236068

29

1860498

1.618034

2.618034

4.236068

30

3010349

1.618034

2.618034

4.236068

 Table 2: The Lucas sequence and ratios of its terms

The Lucas sequence can be explored by clicking on the Lucas tab in the Excel spreadsheet. Although the initial values are different, we see that the ratios tend to converge to the same values as those of the Fibonacci Sequence.

Theorem: The ratios of successive terms in a Fibonacci sequence converge to the Golden Ratio. The ratios of pairs of second terms converge to the valueand the ratios of pairs of third terms converge to the value.

Proof:

Part 1.  Proof that the ratio of sequential terms converges to.

Assume

If , then .

Thus, , so

Rearranging, we have:

Solving for L with the quadratic formula, we have:

If , then L<0, which is not possible, so we discard this root.

Thus,

.                      

Part 2: Proof that the ratios of second terms converge to .

and since,  from part 1:

Part 3: Proof that the ratios of third terms converge to.

Since (from part 2) and since (from part 1), we have:

Thus,

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