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 | 1.66666666666667 |
| 5 | 1.6 |
| 8 | 1.625 |
| 13 | 1.61538461538462 |
| 21 | 1.61904761904762 |
| 34 | 1.61764705882353 |
| 55 | 1.61818181818182 |
| 89 | 1.69797752808989 |
| 144 | 1.61805555555556 |
| 233 | 1.61802575107296 |
| 377 | 1.61803713527851 |
| 610 | 1.61803278688525 |
| 987 | 1.61803444782168 |
| 1597 | 1.61803381340013 |
| 2584 | 1.61803045572755 |
| 4181 | 1.61803396316671 |
| 6765 | 1.6180339985218 |
| 10946 | 1.61803398501736 |
| 17711 | 1.6180339901756 |
| 28657 | 1.61803398820532 |
| 46368 | 1.6180339889579 |
| 75025 | 1.61803398867044 |
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 | 2.66666666666667 |
| 5 | 2.6 |
| 8 | 2.65 |
| 13 | 2.61538461538462 |
| 21 | 2.61904761904762 |
| 34 | 2.61764705882353 |
| 55 | 2.61818181818182 |
| 89 | 2.61797752808989 |
| 144 | 2.61805555555556 |
| 233 | 2.61802575107296 |
| 377 | 2.61803713527851 |
| 610 | 2.61803278688525 |
| 987 | 2.61803444782168 |
| 1597 | 2.61803405572755 |
| 2584 | 2.61803396316671 |
| 4181 | 2.6180339985218 |
| 6765 | 2.61803398501736 |
| 10946 | 2.6180339901756 |
| 17711 | 2.61803398820533 |
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 | 4.33333333333333 |
| 5 | 4.2 |
| 8 | 4.25 |
| 13 | 4.23076923076923 |
| 21 | 4.23809523809524 |
| 34 | 4.23529411764706 |
| 55 | 4.23636363636364 |
| 89 | 4.23595505617978 |
| 144 | 4.23611111111111 |
| 233 | 4.23605150214592 |
| 377 | 4.23607427055703 |
| 610 | 4.23606557377049 |
| 987 | 4.23606889564336 |
| 1597 | 4.23606762680025 |
| 2584 | 4.23606811145511 |
| 4181 | 4.23606792633341 |
| 6765 | 4.23606799704361 |
| 10946 | 4.23606797003472 |
| 17711 | 4.23606798035119 |
| 28657 | 4.23606797751065 |
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.