Assignment 12 Writeup

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.