﻿ Fun with Fibonacci Numbers

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?

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