Investigating a Fibonnaci Sequence

by

Kimberly Burrell

First, we must generate a Fibonnaci Sequence using f(0) = 1, f(1) = 1, and f(n) = f(n-1) + f(n-2).

 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025

We now construct the ratio of each pair of adjacent terms in the Fibonnaci Sequence. This ratio is shown in the second column.

 1 1 1 2 2 1.5 3 1.66667 5 1.6 8 1.625 13 1.61538 21 1.61905 34 1.61765 55 1.61818 89 1.69798 144 1.61806 233 1.61803 377 1.61804 610 1.61803 987 1.61803 1597 1.61803 2584 1.61803 4181 1.61803 6765 1.61803 10946 1.61803 17711 1.61803 28657 1.61803 46368 1.61803 75025 1.61803

One can observe that as n increases the ratio of adjacent terms is approaching the Golden Ratio, which is 1.6180339887. To prove this finding, click here.

Now, let's investigate the ratio of every second term of the Fibonnaci Sequence.

 1 2 1 3 2 2.5 3 2.66667 5 2.6 8 2.65 13 2.61538 21 2.61905 34 2.61765 55 2.61818 89 2.61798 144 2.61806 233 2.61803 377 2.61804 610 2.61803 987 2.61803 1597 2.61803 2584 2.61803 4181 2.61803 6765 2.61803 10946 2.61803 17711 2.61803

Here one can observe that the ratio approaches 2.618033988 or the Golden Ratio + 1.

Let's repeat this process once more to find the ratio of every third term in the Fibonnaci Sequence.

 1 3 1 5 2 4 3 4.33333 5 4.2 8 4.25 13 4.23077 21 4.2381 34 4.23529 55 4.23636 89 4.23596 144 4.23611 233 4.23605 377 4.23607 610 4.23607 987 4.23607 1597 4.23607 2584 4.23607 4181 4.23607 6765 4.23607 10946 4.23607 17711 4.23607 28657 4.23607

In this particular case, one can observe that the ratio of every third term of the Fibonnaci Sequence approaches 4.236067977, which is twice the Golden Ratio + 1. To investigate these limits of ratios for different values of f(0) and f(1), click here.

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