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)=1 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 to generate the Fibonnaci sequence from n=0 to n=30 in the table below. For the cell in column A row 1 and column A row 2, I entered a value of 1. For the column A row 3 cell, I entered the formula =a2+a1 to get a value of 2. Next, I took the lower right corner of the cell in column A row 3 and copied the formula all the way down to row 30. Now I can check to make sure my values are correct. For example, choose the value 75025 from a25. Now check this value using the formula f(25)=f(24)+f(23). Does, 46368+28657=75025? Yes it does, so the values are correct.
Notice below that I also constructed the ratio
of each pair of adjacent terms in the Fibonnaci sequence. This
was performed by selecting b2 and entering the formula =a2/a1.
Again I took the lower right corner of cell b2 and copied all
the way down the cell b30. We see that the ratio starts out at
1, increases to 2 and then decreases to 1.5 Then it varies between
1.6 and 1.7 for several values before it appears to stay at 1.618.
We can make the observation that the ratio seems to remain the
same as n gets larger.
| 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 chose some arbitrary integers for f(0) and f(1) other than 1. I decided to make f(0)=3 and f(1)=4. Notice that in the first column, the values are very different from the values of the Fibonnaci sequence in the first table. However, the interesting observation is that the ratio of the adjacent terms in the second column still approach the same limit of l.618 like the Fibonnaci sequence.
| 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 |
| 3010349 | 1.61803398874925 |
Lastly, I looked at a sequence where f(0) = 1 and f(1) = 3 which is called a Lucas Sequence. Notice again that a similar ratio exists such that as n gets large, the ratio stays very close to 1.618.
| 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 |
To conclude, we can say that
the limit of the ratio for the adjacent terms of the Fibonnaci
sequence and the Lucas sequence both approach 1.618, as well as,
any other similar sequence, as n goes to infinity.