In this assignment, we were to use a spreadsheet to do an exploration of some topic in mathematics. I choose to do problem 4, which was to generate a Fibonnaci sequence using a spreadsheet. First, lets discuss what the Fibonnaci sequence is and how to generate it.

** Fibonacci sequence **is a sequence
of numbers where you start off with two numbers called f(0) and
f(1). To find the next number in the sequence, you add f(0)
and f(1). To find the fourth number in the sequence, you
add the third number to f(1), and so on. So each number
after the first two numbers, is the sum of the two previous numbers.
One example is given below.

If f(0)=1 and f(1)=2, then the sequence would be 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc....

Using variables to show the sequence, we can say that f(n) = f(n-1) + f(n-2).

In this assignment, we will look at several examples of the Fibonacci sequence, and we will explore the ratios of the adjacent terms in the Fibonnaci sequence.

In our first example, lets say that f(0)=0 and f(1)=1. We are going to use Microsoft Excel to explore the sequence and the ratios of adjacent terms. Here is the result.

0 | 0 | 0 | 0 | 0 |

1 | 1 | 0.5 | 0.33333333 | 0.2 |

1 | 0.5 | 0.33333333 | 0.2 | 0.125 |

2 | 0.66666667 | 0.4 | 0.25 | 0.15384615 |

3 | 0.6 | 0.375 | 0.23076923 | 0.14285714 |

5 | 0.625 | 0.38461538 | 0.23809524 | 0.14705882 |

8 | 0.61538462 | 0.38095238 | 0.23529412 | 0.14545455 |

13 | 0.61904762 | 0.38235294 | 0.23636364 | 0.14606742 |

21 | 0.61764706 | 0.38181818 | 0.23595506 | 0.14583333 |

34 | 0.61818182 | 0.38202247 | 0.23611111 | 0.14592275 |

55 | 0.61797753 | 0.38194444 | 0.2360515 | 0.14588859 |

89 | 0.61805556 | 0.38197425 | 0.23607427 | 0.14590164 |

144 | 0.61802575 | 0.38196286 | 0.23606557 | 0.14589666 |

233 | 0.61803714 | 0.38196721 | 0.2360689 | 0.14589856 |

377 | 0.61803279 | 0.38196555 | 0.23606763 | 0.14589783 |

610 | 0.61803445 | 0.38196619 | 0.23606811 | 0.14589811 |

987 | 0.61803381 | 0.38196594 | 0.23606793 | 0.145898 |

1597 | 0.61803406 | 0.38196604 | 0.236068 | 0.14589804 |

2584 | 0.61803396 | 0.381966 | 0.23606797 | 0.14589803 |

4181 | 0.618034 | 0.38196601 | 0.23606798 | 0.14589804 |

6765 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

10946 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

17711 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

28657 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

46368 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

75025 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

In the first column, we have the Fibonnaci sequence. In the second column of numbers, we have the ratio of each pair of adjacent terms. In the third column, we have the ratio of every second term. In the fourth column we have the ratio of every third term. In the fifth column we have the ratio of every fourth term.

In the first column, we can see that as f(n) increases its value increases. The numbers get larger very quickly. From f(0) to f(25), we have gone from 0 to 75,025. Therefore, we can see that in the Fibonnacci sequence we have a very rapid rate of increase in the numbers.

In the second column, we have the ratio of the adjacent numbers in the sequence. For example, the first number is the ratio of f(0)/f(1) or 0/1 which is equal to zero. The second number is the ratio f(1)/f(2) or 1/1 which is equal to 1, and so on. We can see that the ratio of adjacent numbers has the limit of 0.61803399. This number is approximately the Golden Ratio which is seen below.

In the third column, we have the ratio of every second term in the sequence. For example, the first term is f(0)/f(2) or 0/1 which equals 0. The second term is f(1)/f(3) or 1/2 which equals 0.5, and so on. This set of ratios also has a limit. The limit here is 0.31896601.

In the fourth column, we have the ratio of every third term in the sequence. For example, the first term is f(0)/f(3) or 0/2 which equals 0. The second term is f(1)/f(4) or 1/3 which equals 0.33333..., and so on. This set of ratios has the limit 0.23606798.

In the fifth column, we have the ratio of every fourth term in the sequence. For example, the first term is f(0)/f(4) or 0/3 which equals 0. The second term is f(1)/f(5) or 1/5 which equals 0.2. This set of ratios has the limit 0.14589803.

In our second example lets use a special Fibonnaci sequence called the Lucas sequence where f(0)=1 and f(1)=3. Lets also look at the ratios of adjacent terms and their limits as well. Here is the result.

1 | 0.33333333 | 0.25 | 0.14285714 | 0.09090909 |

3 | 0.75 | 0.42857143 | 0.27272727 | 0.16666667 |

4 | 0.57142857 | 0.36363636 | 0.22222222 | 0.13793103 |

7 | 0.63636364 | 0.38888889 | 0.24137931 | 0.14893617 |

11 | 0.61111111 | 0.37931034 | 0.23404255 | 0.14473684 |

18 | 0.62068966 | 0.38297872 | 0.23684211 | 0.14634146 |

29 | 0.61702128 | 0.38157895 | 0.23577236 | 0.14572864 |

47 | 0.61842105 | 0.38211382 | 0.2361809 | 0.14596273 |

76 | 0.61788618 | 0.38190955 | 0.23602484 | 0.14587332 |

123 | 0.61809045 | 0.38198758 | 0.23608445 | 0.14590747 |

199 | 0.61801242 | 0.38195777 | 0.23606168 | 0.14589443 |

322 | 0.61804223 | 0.38196916 | 0.23607038 | 0.14589941 |

521 | 0.61803084 | 0.38196481 | 0.23606706 | 0.14589751 |

843 | 0.61803519 | 0.38196647 | 0.23606833 | 0.14589823 |

1364 | 0.61803353 | 0.38196584 | 0.23606784 | 0.14589796 |

2207 | 0.61803416 | 0.38196608 | 0.23606803 | 0.14589806 |

3571 | 0.61803392 | 0.38196599 | 0.23606796 | 0.14589802 |

5778 | 0.61803401 | 0.38196602 | 0.23606798 | 0.14589804 |

9349 | 0.61803398 | 0.38196601 | 0.23606797 | 0.14589803 |

15127 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

In our first column, we have the Fibonnaci sequence of numbers. As you can see in this example, the set of numbers also rapidly increase. Next lets look at the ratios of the adjacent terms. In the second column, we have the ratio of each pair of adjacent terms. We can see that the limit is the same in this example as in the first one, 0.61803399. In the third column, we have the ratio of every second term. In this set of ratios we also have the same limit as before, 0.38196601. In the fourth column, we have the ratio of every third term. We also have the same limit as before, 0.23606798. In the fifth column, we have the ratio of every fourth term. We also have the same limit as before, 0.14589803.

In the next example, lets use f(0)=1 and f(1)=4 and see what happens to the sequence and the ratio of terms.

1 | 0.25 | 0.2 | 0.11111111 | 0.07142857 |

4 | 0.8 | 0.44444444 | 0.28571429 | 0.17391304 |

5 | 0.55555556 | 0.35714286 | 0.2173913 | 0.13513514 |

9 | 0.64285714 | 0.39130435 | 0.24324324 | 0.15 |

14 | 0.60869565 | 0.37837838 | 0.23333333 | 0.1443299 |

23 | 0.62162162 | 0.38333333 | 0.2371134 | 0.14649682 |

37 | 0.61666667 | 0.3814433 | 0.23566879 | 0.14566929 |

60 | 0.6185567 | 0.38216561 | 0.23622047 | 0.1459854 |

97 | 0.61783439 | 0.38188976 | 0.23600973 | 0.14586466 |

157 | 0.61811024 | 0.38199513 | 0.23609023 | 0.14591078 |

254 | 0.61800487 | 0.38195489 | 0.23605948 | 0.14589316 |

411 | 0.61804511 | 0.38197026 | 0.23607122 | 0.14589989 |

665 | 0.61802974 | 0.38196439 | 0.23606674 | 0.14589732 |

1076 | 0.61803561 | 0.38196663 | 0.23606845 | 0.14589831 |

1741 | 0.61803337 | 0.38196577 | 0.2360678 | 0.14589793 |

2817 | 0.61803423 | 0.3819661 | 0.23606805 | 0.14589807 |

4558 | 0.6180339 | 0.38196598 | 0.23606795 | 0.14589802 |

7375 | 0.61803402 | 0.38196602 | 0.23606799 | 0.14589804 |

11933 | 0.61803398 | 0.38196601 | 0.23606797 | 0.14589803 |

19308 | 0.61803399 | 0.38196601 | 0.23606798 | 0.14589803 |

In the first column, we can see the sequence of numbers increases rapidly again. We can also see that the limits of the ratios of the next four columns are the same as they were in the first two examples. Therefore, we can make the conjecture that the limits of the ratios will always be the same for any Fibonnaci sequence. The limit of the ratio of every pair of adjacent term will always be 0.61803399. The limit of every second term will always be 0.38196601. The limit of every third term will always be 0.23606798. The limit of every fourth term will always be 0.14589803.

We could look at many other examples of the Fibonacci sequence for different f(0) and f(1) and get the same results.