Assignment 12
Investigation 3
Exploring the Fibonacci
Sequence
First we will generate a Fibonacci
sequence in the first column of our spreadsheet using f(0) = 1,
f(1) = 1, and f(n) = f(n-1) + f(n-2)
| n Value |
Fibonnaci Value |
| 0 |
1 |
| 1 |
1 |
| 2 |
2 |
| 3 |
3 |
| 4 |
5 |
| 5 |
8 |
| 6 |
13 |
| 7 |
21 |
| 8 |
34 |
| 9 |
55 |
| 10 |
89 |
| 11 |
144 |
| 12 |
233 |
| 13 |
377 |
| 14 |
610 |
| 15 |
987 |
| 16 |
1597 |
| 17 |
2584 |
| 18 |
4181 |
| 19 |
6765 |
| 20 |
10946 |
Now if we create a third
column with the ratio of each pair of adjacent terms in the Fibonacci
sequence.
| n Value |
Fibonnaci Value |
Ratio |
| 0 |
1 |
1 |
| 1 |
1 |
1 |
| 2 |
2 |
2 |
| 3 |
3 |
1.5 |
| 4 |
5 |
1.66666666666667 |
| 5 |
8 |
1.6 |
| 6 |
13 |
1.625 |
| 7 |
21 |
1.61538461538462 |
| 8 |
34 |
1.61904761904762 |
| 9 |
55 |
1.61764705882353 |
| 10 |
89 |
1.61818181818182 |
| 11 |
144 |
1.61797752808989 |
| 12 |
233 |
1.61805555555556 |
| 13 |
377 |
1.61802575107296 |
| 14 |
610 |
1.61803713527851 |
| 15 |
987 |
1.61803278688525 |
| 16 |
1597 |
1.61803444782168 |
| 17 |
2584 |
1.61803381340013 |
| 18 |
4181 |
1.61803405572755 |
| 19 |
6765 |
1.61803396316671 |
| 20 |
10946 |
1.6180339985218 |
It is obvious that the
ratios are approaching 1.618 or something close to 1.618. Now
lets do the same thing with the ration of every second term.
| n Value |
Fibonnaci Value |
Adjacent Ratio |
Second Ratio |
| 0 |
1 |
1 |
|
| 1 |
1 |
1 |
|
| 2 |
2 |
2 |
2 |
| 3 |
3 |
1.5 |
3 |
| 4 |
5 |
1.66666666666667 |
2.5 |
| 5 |
8 |
1.6 |
2.66666666666667 |
| 6 |
13 |
1.625 |
2.6 |
| 7 |
21 |
1.61538461538462 |
2.625 |
| 8 |
34 |
1.61904761904762 |
2.61538461538462 |
| 9 |
55 |
1.61764705882353 |
2.61904761904762 |
| 10 |
89 |
1.61818181818182 |
2.61764705882353 |
| 11 |
144 |
1.61797752808989 |
2.61818181818182 |
| 12 |
233 |
1.61805555555556 |
2.61797752808989 |
| 13 |
377 |
1.61802575107296 |
2.61805555555556 |
| 14 |
610 |
1.61803713527851 |
2.61802575107296 |
| 15 |
987 |
1.61803278688525 |
2.61803713527851 |
| 16 |
1597 |
1.61803444782168 |
2.61803278688525 |
| 17 |
2584 |
1.61803381340013 |
2.61803444782168 |
| 18 |
4181 |
1.61803405572755 |
2.61803381340013 |
| 19 |
6765 |
1.61803396316671 |
2.61803405572755 |
| 20 |
10946 |
1.6180339985218 |
2.61803396316671 |
| 21 |
17711 |
1.61803398501736 |
2.6180339985218 |
| 22 |
28657 |
1.6180339901756 |
2.61803398501736 |
| 23 |
46368 |
1.61803398820532 |
2.6180339901756 |
| 24 |
75025 |
1.6180339889579 |
2.61803398820533 |
| 25 |
121393 |
1.61803398867044 |
2.6180339889579 |
What would happen to
our ratios if we used two different values for f(0) and f(1).
For example, like the Lucas sequence let f(0)=1 and f(1)=3.
| n Value |
Fibonnaci Value |
Adjacent Ratio |
Second Ratio |
| 0 |
1 |
0.333333333333333 |
|
| 1 |
3 |
3 |
|
| 2 |
4 |
1.33333333333333 |
4 |
| 3 |
7 |
1.75 |
2.33333333333333 |
| 4 |
11 |
1.57142857142857 |
2.75 |
| 5 |
18 |
1.63636363636364 |
2.57142857142857 |
| 6 |
29 |
1.61111111111111 |
2.63636363636364 |
| 7 |
47 |
1.62068965517241 |
2.61111111111111 |
| 8 |
76 |
1.61702127659574 |
2.62068965517241 |
| 9 |
123 |
1.61842105263158 |
2.61702127659574 |
| 10 |
199 |
1.61788617886179 |
2.61842105263158 |
| 11 |
322 |
1.61809045226131 |
2.61788617886179 |
| 12 |
521 |
1.61801242236025 |
2.61809045226131 |
| 13 |
843 |
1.61804222648752 |
2.61801242236025 |
| 14 |
1364 |
1.61803084223013 |
2.61804222648752 |
| 15 |
2207 |
1.61803519061584 |
2.61803084223013 |
| 16 |
3571 |
1.6180335296783 |
2.61803519061584 |
| 17 |
5778 |
1.61803416409969 |
2.6180335296783 |
| 18 |
9349 |
1.61803392177224 |
2.61803416409969 |
| 19 |
15127 |
1.61803401433308 |
2.61803392177224 |
| 20 |
24476 |
1.61803397897799 |
2.61803401433308 |
| 21 |
39603 |
1.61803399248243 |
2.61803397897799 |
| 22 |
64079 |
1.61803398732419 |
2.61803399248243 |
| 23 |
103682 |
1.61803398929446 |
2.61803398732419 |
| 24 |
167761 |
1.61803398854189 |
2.61803398929446 |
| 25 |
271443 |
1.61803398882935 |
2.61803398854189 |
We can see that our ratios
still approach the same values as the original Fibonacci sequence.
Return
to Assignment 12