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 1.66666666666667
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

 

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 4.33333333333333
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.

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