Final Project

Part 2

The Fibonacci Sequence


The spreadsheet is a utility tool that can be adapted to many different explorations, presentations, and simulations in mathematics. An essential feature should be the ability to make graphs and charts from the matrix of data. Generate the Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1, f(n) = f(n-1) + f(n-2).


The Fibonacci sequence of numbers has been studied for hundreds of years for his mathematical beauty and its appearance in nature and the man-made world. This investigation will begin with the basics of generating the Fibonacci sequence and continue to explore the golden ration, the Lucas sequence and the existence of the sequence in the world around us.

Let's begin with a bit of history of the number sequence.The sequence was named for the mathematician known as Leonardo of Pisa or Fibonacci. He lived in the 1400's in Italy, but traveled many places learning about math, wherever he went. He is credited with 'discovering' the number sequence named for him, but it is difficult to determine if he was the first to ever study it.

The formula to generate the Fibonacci sequence is as follows:

 

f(0) = 1

f(1) = 1

f(2) = f(1) + f(0) = 1 + 1 = 2

f(3) = f(2) + f(1) = 2 + 1 = 3

f(4) = f(3) + f(2) = 3 + 2 = 5

.

.

.

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

Generating this sequence by hand is beneficial, yet a spreadsheet can help to continue the sequence for larger terms and also help to manipulate in order to recognize any relationships.

The following is the first 50 terms of the Fibonacci sequence.

Term

Fibonacci Number
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
13 233
14 377
15 610
16 987
17 1597
18 2584
19 4181
20 6765
21 10946
22 17711
23 28657
24 46368
25 75025
26 121393
27 196418
28 317811
29 514229
30 832040
31 1346269
32 2178309
33 3524578
34 5702887
35 9227465
36 14930352
37 24157817
38 39088169
39 63245986
40 102334155
41 165580141
42 267914296
43 433494437
44 701408733
45 1134903170
46 1836311903
47 2971215073
48 4807526976
49 7778742049
50 12586269025

It is interesting to note that Fibonacci numbers, such as 2,3,5,8,13, 21,...have been found in many facets of nature. Some examples include the petals and stems of flowers; the number of seeds in a sunflower; the leaves in pine cones, pineapples and artichokes; and the number of sections in a grapefruit.

The Fibonacci sequence also generates the golden ratio. This ratio can be created by dividing successive numbers of the sequence. The ratio appears as n increases.

The golden ratio is equal to approximately 1.618....It is a repeating decimal with no known pattern. Another way to write the golden ratio is .

The spreadsheet is able to calculate the ratio for any number of terms.

Term

Ratio
1
2 1
3 2
4 1.5
5 1.66666666666667
6 1.6
7 1.625
8 1.61538461538462
9 1.61904761904762
10 1.61764705882353
11 1.61818181818182
12 1.61797752808989
13 1.61805555555556
14 1.61802575107296
15 1.61803713527851
16 1.61803278688525
17 1.61803444782168
18 1.61803381340013
19 1.61803405572755
20 1.61803396316671
21 1.6180339985218
22 1.61803398501736
23 1.6180339901756
24 1.61803398820532
25 1.6180339889579
26 1.61803398867044
27 1.61803398878024
28 1.6180339887383
29 1.61803398875432
30 1.6180339887482

It might be helpful to see how each successive numbers tend toward the golden ratio as n increase with a graph.


There are other sequences of numbers that follow the generating method of the Fibonacci sequence. The sequence begins with f(0) = 2. The following table has only the first 15 terms (to save space).

Term

Sequence starting with 2
1 2
2 2
3 4
4 6
5 10
6 16
7 26
8 42
9 68
10 110
11 178
12 288
13 466
14 754
15 1220

The ratio, however, between successive numbers is the same. As n increases the ratio tends towards the golden ratio. In fact, the list of ratios is an exact replica of the ratios of the Fibonacci sequence.

Sequence starting with 2

Ratio
2
2 1
4 2
6 1.5
10 1.66666666666667
16 1.6
26 1.625
42 1.61538461538462
68 1.61904761904762
110 1.61764705882353
178 1.61818181818182
288 1.61797752808989
466 1.61805555555556
754 1.61802575107296
1220 1.61803713527851
1974 1.61803278688525
3194 1.61803444782168
5168 1.61803381340013
8362 1.61803405572755
13530 1.61803396316671
21892 1.6180339985218
35422 1.61803398501736
57314 1.6180339901756

Looking at various number sequences such as adding each successive odd numbers (starting with the first two numbers) together produces the same result when determining their ratios.

Term

Odd

Ratio
1 1
2 3 3
3 4 1.33333333333333
4 7 1.75
5 11 1.57142857142857
6 18 1.63636363636364
7 29 1.61111111111111
8 47 1.62068965517241
9 76 1.61702127659574
10 123 1.61842105263158
11 199 1.61788617886179
12 322 1.61809045226131
13 521 1.61801242236025
14 843 1.61804222648752
15 1364 1.61803084223013
16 2207 1.61803519061584
17 3571 1.6180335296783
18 5778 1.61803416409969
19 9349 1.61803392177224
20 15127 1.61803401433308
21 24476 1.61803397897799
22 39603 1.61803399248243
23 64079 1.61803398732419
24 103682 1.61803398929446

Adding the even numbers (starting with the first two numbers) together produces this list of numbers

Term

Even

Ratio
1 2
2 4 2
3 6 1.5
4 10 1.66666666666667
5 16 1.6
6 26 1.625
7 42 1.61538461538462
8 68 1.61904761904762
9 110 1.61764705882353
10 178 1.61818181818182
11 288 1.61797752808989
12 466 1.61805555555556
13 754 1.61802575107296
14 1220 1.61803713527851
15 1974 1.61803278688525
16 3194 1.61803444782168
17 5168 1.61803381340013
18 8362 1.61803405572755
19 13530 1.61803396316671
20 21892 1.6180339985218
21 35422 1.61803398501736
22 57314 1.6180339901756
23 92736 1.61803398820532
24 150050 1.6180339889579
25 242786 1.61803398867044

 

Another ratio to investigate is the ratio of every second term. For space, I have placed only the table of the Fibonacci numbers below. Observe any similarities to the ratio of each successive numbers.

Fibonacci Number

Ratio of 2nd Terms
1
1
2 2
3 3
5 2.5
8 2.66666666666667
13 2.6
21 2.625
34 2.61538461538462
55 2.61904761904762
89 2.61764705882353
144 2.61818181818182
233 2.61797752808989
377 2.61805555555556
610 2.61802575107296
987 2.61803713527851
1597 2.61803278688525
2584 2.61803444782168
4181 2.61803381340013

The ratio of each successive term tends toward the number 2.618...Notice that it is similar to the golden ratio. The relationship between these two numbers (1.618) and (2.1618) is extraodinary. It is easy to see that they have an arithmetic difference of 1, but 2.618 is also the square of 1.618. Thus, you can say that the golden ratio's square is equal to the golden ratio plus 1.

Fibonacci Number

Ratio of 2nd Terms

Ratio of 3rd Terms

Ratio of 4th Terms

Ratio of 5th Terms
1
1
2 2 3 5 8
3 3 5 8 13
5 2.5 4 6.5 10.5
8 2.66666666666667 4.33333333333333 7 11.3333333333333
13 2.6 4.2 6.8 11
21 2.625 4.25 6.875 11.125
34 2.61538461538462 4.23076923076923 6.84615384615385 11.0769230769231
55 2.61904761904762 4.23809523809524 6.85714285714286 11.0952380952381
89 2.61764705882353 4.23529411764706 6.85294117647059 11.0882352941176
144 2.61818181818182 4.23636363636364 6.85454545454545 11.0909090909091

Notice the last line of the table. This is the value that each ratio tends towards as n increases. It also has the distinction of being the golden ratio squared, cubed, quadrupled and quintupled.

1.61803

2.6180210809

4.23603664952863

6.85403438003681

11.090033247931

The use of spreadsheets can peak interest into the patterns and mathematical basis for the golden ratio. The discussion that follows is a nice connection back to the Fibonacci sequence.

As mentioned above, the golden ratio has the distinction of having its square equal to itself plus one. The support for this fact can be found when we look at a line segment that has been parted into the golden ratio.

Any line segment that has been divided into the golden segment can be written as the proportion

If we substitute

We find

.

Multiplying each side by x, gives us

Again, this supports the calculations in our table. The square of the golden ratio is equal to itself plus 1.

To connect this finding to the Fibonacci sequence continue to find the "values" of the powers of x (assuming that x is representative of the golden ratio).

 

 

 

 

 

Notice that the coefficients in each term are successive Fibonacci numbers. This patterns continues and is consistent for each power of x.

More information on Fibonacci and the Golden Ratio can be found on various web sites.


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