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.