Exploring a Fibonnaci Sequence

by Amy Benson

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

 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418

Now, we construct the ratio of each pair of adjacent terms in the Fibonnaci sequence. (Shown in the second column.)

 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 1.61798 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 121393 1.61803 196418 1.61803 317811 1.61803 514229 1.61803 832040 1.61803 1346269 1.61803 2178309 1.61803 3524578 1.61803 5702887 1.61803 9227465 1.61803 14930352 1.61803 24157817 1.61803

Notice, that as n increases, the ratio of adjacent terms approaches 1.618033988738303 (the Golden Ratio).

Next, let's investigate the ratio of every second term of the Fibonnaci sequence.

 1 3 2 2.5 3 2.66667 5 2.6 8 2.625 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 28657 2.61803 46368 2.61803 75025 2.61803 121393 2.61803 196418 2.61803 317811 2.61803 514229 2.61803 832040 2.61803 1346269 2.61803 2178309 2.61803 3524578 2.61803 5702887 2.61803 9227465 2.61803 14930352 2.61803

In this case, the ratio approaches 2.618033988 (the Golden Ratio + 1).

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

 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 46368 4.23607 75025 4.23607 121393 4.23607 196418 4.23607 317811 4.23607 514229 4.23607 832040 4.23607 1346269 4.23607 2178309 4.23607 3524578 4.23607 5702887 4.23607 9227465 4.23607

In this case, we see that the ratio of every third term of the Fibonnaci Sequence approaches 4.236067977499 (twice the Golden Ratio + 1 or twice the ratio of every second term of the Fibonnaci Sequence minus 1).

To investigate these limits of ratios for different values of f(0) and f(1), click here.