Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1 and f(n) = f(n-1) + f(n-2)
a. Construct the ratio of each pair of adjacent terms in the Fibonnaci sequence. What happens as n increases? What about the ratio of every second term? etc.
b. Explore sequences where f(0) and f(1) are some arbitrary integers other than 1. If f(0) and f(1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.
I used Microsoft Excel's
spreadsheet to
construct the Fibonnaci sequence from n=0 to n=30. Below is the
result f(30). Notice how each term is the sum of the previous
two terms such that 1+1 = 2 and 2+1 = 3, etc.
| 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 |
| 121393 |
| 196418 |
| 317811 |
| 514229 |
| 832040 |
Next, I constructed the ratio of each pair of adjacent terms in the Fibonnaci sequence.
Notice that as n increases, the ratio starts out at 1 and then jumps up to 2 and back down to 1.5.
Then it jumps a little over 1.6 and then back down to exactly 1.6.
Then, as n is over 5 or 6, it seems that the ratio remains very close to 1.618. That is neat how the ratio seems to remain the same as n gets large.
| 1 | |
| 1 | 1 |
| 2 | 2 |
| 3 | 1.5 |
| 5 | 1.66666666666667 |
| 8 | 1.6 |
| 13 | 1.625 |
| 21 | 1.61538461538462 |
| 34 | 1.61904761904762 |
| 55 | 1.61764705882353 |
| 89 | 1.61818181818182 |
| 144 | 1.61797752808989 |
| 233 | 1.61805555555556 |
| 377 | 1.61802575107296 |
| 610 | 1.61803713527851 |
| 987 | 1.61803278688525 |
| 1597 | 1.61803444782168 |
| 2584 | 1.61803381340013 |
| 4181 | 1.61803405572755 |
| 6765 | 1.61803396316671 |
| 10946 | 1.6180339985218 |
| 17711 | 1.61803398501736 |
| 28657 | 1.6180339901756 |
| 46368 | 1.61803398820532 |
| 75025 | 1.6180339889579 |
| 121393 | 1.61803398867044 |
| 196418 | 1.61803398878024 |
| 317811 | 1.6180339887383 |
| 514229 | 1.61803398875432 |
| 832040 | 1.6180339887482 |
Next, I picked f(0) and f(1) to be some arbitrary integers other than 1. I picked f(0)=2 and f(1)=3. Notice how this is the same Fibonnaci sequence I began with. It just starts at a later point. After taking a look at the ratio of the adjacent terms, it would make sense that the limit of the ratios approaches the same 1.618 as the Fibonnaci sequence did since they are the same thing!
| 2 | |
| 3 | 1.5 |
| 5 | 1.66666666666667 |
| 8 | 1.6 |
| 13 | 1.625 |
| 21 | 1.61538461538462 |
| 34 | 1.61904761904762 |
| 55 | 1.61764705882353 |
| 89 | 1.61818181818182 |
| 144 | 1.61797752808989 |
| 233 | 1.61805555555556 |
| 377 | 1.61802575107296 |
| 610 | 1.61803713527851 |
| 987 | 1.61803278688525 |
| 1597 | 1.61803444782168 |
| 2584 | 1.61803381340013 |
| 4181 | 1.61803405572755 |
| 6765 | 1.61803396316671 |
| 10946 | 1.6180339985218 |
| 17711 | 1.61803398501736 |
| 28657 | 1.6180339901756 |
| 46368 | 1.61803398820532 |
| 75025 | 1.6180339889579 |
| 121393 | 1.61803398867044 |
| 196418 | 1.61803398878024 |
| 317811 | 1.6180339887383 |
| 514229 | 1.61803398875432 |
| 832040 | 1.6180339887482 |
| 1346269 | 1.61803398875054 |
| 2178309 | 1.61803398874965 |
Here, I looked at sequences where f(0) = 1 and f(1) = 3. This is called the Lucas Sequence. See the spreadsheet below.
| 1 |
| 3 |
| 4 |
| 7 |
| 11 |
| 18 |
| 29 |
| 47 |
| 76 |
| 123 |
| 199 |
| 322 |
| 521 |
| 843 |
| 1364 |
| 2207 |
| 3571 |
| 5778 |
| 9349 |
| 15127 |
| 24476 |
| 39603 |
| 64079 |
| 103682 |
| 167761 |
| 271443 |
| 439204 |
| 710647 |
| 1149851 |
| 1860498 |
Now, let's look at the ratio between the two adjacent terms. It appears that the same sort of ratio exists such that as n gets large, the ratio stays very close to 1.618.
The ratio starts at 3 and then jumps down to 1.333 and then up and down until it starts approaching 1.618.
I guess it could be said that the limit of the ratio for the adjacent terms of the Fibonnaci sequence and the Lucas sequence both approach 1.618 as n goes to infinity.
| 1 | |
| 3 | 3 |
| 4 | 1.33333333333333 |
| 7 | 1.75 |
| 11 | 1.57142857142857 |
| 18 | 1.63636363636364 |
| 29 | 1.61111111111111 |
| 47 | 1.62068965517241 |
| 76 | 1.61702127659574 |
| 123 | 1.61842105263158 |
| 199 | 1.61788617886179 |
| 322 | 1.61809045226131 |
| 521 | 1.61801242236025 |
| 843 | 1.61804222648752 |
| 1364 | 1.61803084223013 |
| 2207 | 1.61803519061584 |
| 3571 | 1.6180335296783 |
| 5778 | 1.61803416409969 |
| 9349 | 1.61803392177224 |
| 15127 | 1.61803401433308 |
| 24476 | 1.61803397897799 |
| 39603 | 1.61803399248243 |
| 64079 | 1.61803398732419 |
| 103682 | 1.61803398929446 |
| 167761 | 1.61803398854189 |
| 271443 | 1.61803398882935 |
| 439204 | 1.61803398871955 |
| 710647 | 1.61803398876149 |
| 1149851 | 1.61803398874547 |
| 1860498 | 1.61803398875159 |
In conclusion, it appears that all such sequences have the same limit of the ratio of successive terms (being 1.618).
This concludes my investigation of Fibonnaci sequences using a spreadsheet.