Final Project 1: Fabulous Fibonacci Numbers

by Elizabeth Gieseking


Leonardo of Pisa (c. 1170-1250), commonly known as Fibonacci, posed and solved the following problem involving an idealized rabbit population in his book Liber Abaci.

Suppose a newly-born pair of rabbits, one male and one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year? 

https://c2.staticflickr.com/6/5190/5636053756_653b220d6f_z.jpg

The table below illustrates what is happening.  When the rabbits are first placed in the field, their age is 0 months.  At this point they are immature and cannot mate, so at the end of month 1, there is still one pair of rabbits.  At that point they are mature and mate, so at the end of the second month there are 2 pairs of rabbits, one mature and one immature.  In each month after the first one, the number of immature rabbit pairs is equal to the number of mature rabbit pairs in the previous month.  Likewise the number of mature rabbit pairs is equal to the total number of rabbit pairs in the previous month.

 

Month

Mature Rabbit

Pairs

Immature Rabbit

Pairs

Total Rabbit Pairs

0

0

1

1

1

1

0

1

2

1

1

2

3

2

1

3

4

3

2

5

5

5

3

8

6

8

5

13

7

13

8

21

8

21

13

34

9

34

21

55

10

55

34

89

11

89

55

144

12

144

89

233

13

233

144

377

14

377

233

610

15

610

377

987

16

987

610

1597

17

1597

987

2584

18

2584

1597

4181

19

4181

2584

6765

20

6765

4181

10946

21

10946

6765

17711

22

17711

10946

28657

23

28657

17711

46368

24

46368

28657

75025

25

75025

46368

121393

The resulting sequence of number of rabbit pairs is the famous Fibonacci sequence.  The Fibonacci sequence is defined recursively as: .  By varying the starting values, we can create many similar sequences.  The most famous of these is the Lucas sequence in which .  Another variation is to the change the number of values being summed at each step.  For example, in a tribonacci sequence the three previous terms are summed at each step: .

A spreadsheet can be a useful tool for studying the Fibonacci sequence and other related sequences.  We simply enter the starting values and our formula and let the computer do the computation.  This is especially useful if we wish to examine ratios of Fibonacci numbers.  In the following table, we are examining the ratios of a Fibonacci number to a previous Fibonacci number.  In the first ratio, we are calculating , the ratio of the Fibonacci number to the previous term.  In the second ratio, we are calculating , the ratio of the Fibonacci number to the number two spots earlier in the sequence.  Likewise, in the third ratio, we are using terms three apart, and so on.

 

n

0

1

1

1

1

2

2

2

2

3

3

1.5

3

3

4

5

1.6666667

2.5

5

5

5

8

1.6000000

2.6666667

4

8

8

6

13

1.6250000

2.6000000

4.3333333

6.5

13

7

21

1.6153846

2.6250000

4.2000000

7

10.5

8

34

1.6190476

2.6153846

4.2500000

6.8

11.3333333

9

55

1.6176471

2.6190476

4.2307692

6.875

11

10

89

1.6181818

2.6176471

4.2380952

6.8461538

11.125

11

144

1.6179775

2.6181818

4.2352941

6.8571429

11.076923

12

233

1.6180556

2.6179775

4.2363636

6.8529412

11.095238

13

377

1.6180258

2.6180556

4.2359551

6.8545455

11.088235

14

610

1.6180371

2.6180258

4.2361111

6.8539326

11.090909

15

987

1.6180328

2.6180371

4.2360515

6.8541667

11.089887

16

1597

1.6180344

2.6180328

4.2360743

6.8540773

11.090277

17

2584

1.6180338

2.6180344

4.2360656

6.8541114

11.090128

18

4181

1.6180341

2.6180338

4.2360689

6.8540984

11.090185

19

6765

1.6180340

2.6180341

4.2360676

6.8541033

11.090163

20

10946

1.6180340

2.6180340

4.2360681

6.8541014

11.090172

21

17711

1.6180340

2.6180340

4.2360679

6.8541022

11.090169

22

28657

1.6180340

2.6180340

4.2360680

6.8541019

11.090170

23

46368

1.6180340

2.6180340

4.2360680

6.8541020

11.090170

24

75025

1.6180340

2.6180340

4.2360680

6.8541020

11.090170

25

121393

1.6180340

2.6180340

4.2360680

6.8541020

11.090170

We see that these values go up and down but converge fairly quickly.  The number for the first ratio looks familiar – it is  or the golden ratio.  The golden ratio is defined as: .  This can be rewritten as:   We can solve this using the quadratic formula to get:

We also quickly see that the second ratio  converges to .  Although the other ratios converge, it is not yet obvious how these values relate to   If we again examine the equations, we see that   Thus our second ratio is .  We can then verify that our third ratio is , our fourth ratio is , and our fifth ratio is

Now we know that the nth ratio converges to , but we do not have a reason yet.  Let’s go back to our equation for  and use it to generate other powers of .

We see two relationships between the powers of  and the Fibonacci series.

1.   The sequence of powers of  has the same structure as the Fibonacci sequence. 
Just as
, we see that

2.   When we express  as a function of , the coefficients are the Fibonacci numbers.

Interestingly, no matter what values we choose for  and  , the nth ratio will still converge to   The table below shows the Lucas numbers in which .

n

0

1

1

3

3

2

4

1.3333333

4

3

7

1.7500000

2.3333333

7

4

11

1.5714286

2.7500000

3.6666667

11

5

18

1.6363636

2.5714286

4.5000000

6

18

6

29

1.6111111

2.6363636

4.1428571

7.25

9.666667

7

47

1.6206897

2.6111111

4.2727273

6.7142857

11.750000

8

76

1.6170213

2.6206897

4.2222222

6.9090909

10.857143

9

123

1.6184211

2.6170213

4.2413793

6.8333333

11.181818

10

199

1.6178862

2.6184211

4.2340426

6.8620690

11.055556

11

322

1.6180905

2.6178862

4.2368421

6.8510638

11.103448

12

521

1.6180124

2.6180905

4.2357724

6.8552632

11.085106

13

843

1.6180422

2.6180124

4.2361809

6.8536585

11.092105

14

1364

1.6180308

2.6180422

4.2360248

6.8542714

11.089431

15

2207

1.6180352

2.6180308

4.2360845

6.8540373

11.090452

16

3571

1.6180335

2.6180352

4.2360617

6.8541267

11.090062

17

5778

1.6180342

2.6180335

4.2360704

6.8540925

11.090211

18

9349

1.6180339

2.6180342

4.2360671