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 | 1.66666666666667 |
| 5 | 1.6 |
| 8 | 1.625 |
| 13 | 1.61538461538462 |
| 21 | 1.61904761904762 |
| 34 | 1.61764705882353 |
| 55 | 1.61818181818182 |
| 89 | 1.61797752808989 |
| 144 | 1.61805555555556 |
| 233 | 1.61802575107296 |
| 377 | 1.61803713527851 |
| 610 | 1.61803278688525 |
| 987 | 1.61803444782168 |
| 1597 | 1.61803381340013 |
| 2584 | 1.61803405572755 |
| 4181 | 1.61803396316671 |
| 6765 | 1.6180339985218 |
| 10946 | 1.61803398501736 |
| 17711 | 1.6180339901756 |
| 28657 | 1.61803398820532 |
| 46368 | 1.6180339889579 |
| 75025 | 1.61803398867044 |
| 121393 | 1.61803398878024 |
| 196418 | 1.6180339887383 |
| 317811 | 1.61803398875432 |
| 514229 | 1.6180339887482 |
| 832040 | 1.61803398875054 |
| 1346269 | 1.61803398874965 |
| 2178309 | 1.61803398874999 |
| 3524578 | 1.61803398874986 |
| 5702887 | 1.61803398874991 |
| 9227465 | 1.61803398874989 |
| 14930352 | 1.6180339887499 |
| 24157817 | 1.61803398874989 |
Notice, that as n increases, the ratio of adjacent terms approaches 1.618033988738303 (the Golden Ratio).
To prove this finding, click here.
Next, let's investigate the ratio of every second term of the Fibonnaci sequence.
| 1 | 3 |
| 2 | 2.5 |
| 3 | 2.66666666666667 |
| 5 | 2.6 |
| 8 | 2.625 |
| 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 | 2.61803381340013 |
| 2584 | 2.61803405572755 |
| 4181 | 2.61803396316671 |
| 6765 | 2.6180339985218 |
| 10946 | 2.61803398501736 |
| 17711 | 2.6180339901756 |
| 28657 | 2.61803398820533 |
| 46368 | 2.6180339889579 |
| 75025 | 2.61803398867044 |
| 121393 | 2.61803398878024 |
| 196418 | 2.6180339887383 |
| 317811 | 2.61803398875432 |
| 514229 | 2.6180339887482 |
| 832040 | 2.61803398875054 |
| 1346269 | 2.61803398874965 |
| 2178309 | 2.61803398874999 |
| 3524578 | 2.61803398874986 |
| 5702887 | 2.61803398874991 |
| 9227465 | 2.61803398874989 |
| 14930352 | 2.6180339887499 |
Let's repeat this process to find the ratio of every third term in the Fibonnaci Sequence.
| 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 | 4.23606797641065 |
| 46368 | 4.2360679779158 |
| 75025 | 4.23606797734089 |
| 121393 | 4.23606797756049 |
| 196418 | 4.23606797747661 |
| 317811 | 4.23606797750864 |
| 514229 | 4.23606797749641 |
| 832040 | 4.23606797750108 |
| 1346269 | 4.2360679774993 |
| 2178309 | 4.23606797749998 |
| 3524578 | 4.23606797749972 |
| 5702887 | 4.23606797749982 |
| 9227465 | 4.23606797749978 |
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.