Fibonacci Sequence

by

Larousse Charlot

The Fibonacci numbers is defined by the following:

The Fibonacci numbers are named after Leonardo of Pisa, who was also known as Fibonacci.  However, that was not the first time the Fibonacci numbers were discovered.  They were described many years prior to Leonardo in India.

Nonetheless, for our exploration of spreadsheet, we are defining the Fibonacci sequence as follow:

This is the sequence in the spreadsheet of Excel

 n F(n) 0 1 1 1 2 2 3 3 4 5 5 8 6 13 7 21 8 34 9 55 10 89 11 144 12 233 13 377 14 610 15 987 16 1597 17 2584 18 4181 19 6765 20 10946 21 17711 22 28657 23 46368 24 75025 25 121393 26 196418 27 317811 28 514229 29 832040 30 1346269 31 2178309 32 3524578 33 5702887 34 9227465 35 14930352 36 24157817 37 39088169 38 63245986 39 102334155 40 165580141 41 267914296 42 433494437 43 701408733 44 1134903170 45 1836311903

If we were to look at the ratio of F(n) to F(n – 1), we would have

 n F(n) F(n)/F(n-1) 0 1 1 1 1 1 2 2 1 3 3 2 4 5 1.5 5 8 1.666666667 6 13 1.6 7 21 1.625 8 34 1.615384615 9 55 1.619047619 10 89 1.617647059 11 144 1.618181818 12 233 1.617977528 13 377 1.618055556 14 610 1.618025751 15 987 1.618037135 16 1597 1.618032787 17 2584 1.618034448 18 4181 1.618033813 19 6765 1.618034056 20 10946 1.618033963 21 17711 1.618033999 22 28657 1.618033985 23 46368 1.61803399 24 75025 1.618033988 25 121393 1.618033989 26 196418 1.618033989 27 317811 1.618033989 28 514229 1.618033989 29 832040 1.618033989 30 1346269 1.618033989 31 2178309 1.618033989 32 3524578 1.618033989 33 5702887 1.618033989 34 9227465 1.618033989 35 14930352 1.618033989 36 24157817 1.618033989 37 39088169 1.618033989 38 63245986 1.618033989 39 102334155 1.618033989 40 165580141 1.618033989 41 267914296 1.618033989 42 433494437 1.618033989 43 701408733 1.618033989 44 1134903170 1.618033989 45 1836311903 1.618033989

As you can observe, as n increases the ratio of F(n) : F(n – 1) approaches the golden ratio (j) number that is defined by

So, the ratio of F(n) : F(n – 1) is bounded by the golden ration.  Hence, we can have a closed formed expression for the Fibonacci numbers.

An equation is said to have a closed form expression if and only if the equation has at least one solution that can be analytically expressed as a bounded number of a certain function, in this case the golden ratio.

Consider the ratio of F(n) : F(n – 2).  I think an addition to the golden ration will occur because F(n – 2) = F(n – 1 – 1).  See if I am right here, I could be wrong.

If redefined our sequence such that we F(0) = 1 and F(1) = 3, we would have the Lucas number, due to Francois Edouard Anatole Lucas, which is defined as

which follow the same path as Fibonacci sequence, and have the same ratio.

Golden Ratio

Suppose we have a line segment that is divided into two no equal parts. If the ratio of the whole segment to the larger part of the segment is equal to the ratio of larger part of the segment to the smaller part of the segment, the ratio is a golden ratio.  Algebraically, we have

from which we obtain j.

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