 ## 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

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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 2.61802 4.23604 6.85403 11.09

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.