Assignment #11

4. Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1,

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.

The following is the fibonacci sequnce demonstrated on excel:

 The Fibonacci Sequence term# term in seq ratio1 ratio2 1 1 2 1 1 3 2 2 2 4 3 1.5 3 5 5 1.66666666666667 2.5 6 8 1.6 2.66666666666667 7 13 1.625 2.6 8 21 1.61538461538462 2.625 9 34 1.61904761904762 2.61538461538462 10 55 1.61764705882353 2.61904761904762 11 89 1.61818181818182 2.61764705882353 12 144 1.61797752808989 2.61818181818182 13 233 1.61805555555556 2.61797752808989 14 377 1.61802575107296 2.61805555555556 15 610 1.61803713527851 2.61802575107296 16 987 1.61803278688525 2.61803713527851 17 1597 1.61803444782168 2.61803278688525 18 2584 1.61803381340013 2.61803444782168 19 4181 1.61803405572755 2.61803381340013 20 6765 1.61803396316671 2.61803405572755 21 10946 1.6180339985218 2.61803396316671 22 17711 1.61803398501736 2.6180339985218 23 28657 1.6180339901756 2.61803398501736 24 46368 1.61803398820532 2.6180339901756 25 75025 1.6180339889579 2.61803398820533 26 121393 1.61803398867044 2.6180339889579 27 196418 1.61803398878024 2.61803398867044 28 317811 1.6180339887383 2.61803398878024 29 514229 1.61803398875432 2.6180339887383 30 832040 1.6180339887482 2.61803398875432 31 1346269 1.61803398875054 2.6180339887482 32 2178309 1.61803398874965 2.61803398875054 33 3524578 1.61803398874999 2.61803398874965 34 5702887 1.61803398874986 2.61803398874999 35 9227465 1.61803398874991 2.61803398874986

The second column displays each term in the sequence. The third column describes the ratio of consecutive terms. The fourth column is the ratio of every other term.

Looking at the list of ratios, I did not see a pattern or any interesting generalization. As the number of terms increase, the ratios level off at about 1.618 for the ratio of consecutive terms and level off around 2.618 for the ratio of every other term.

I graphed the Fibonaccci sequence in Excel and found that the sequence resembles the exponential. We can see from the graph, that the the rate of change greatly increases as n gets larger, by observing the approximate distance between consecutive points as the x values increase.

Above is a graph of the ratios of the consecutive terms of the Fibonacci sequence. We can see that there is an initial jump between the first two points and then an immediate decrease. Around the 9th term, the sequence appears to be constant. We can see this in the excel chart as the terms eventually level off around 1.618.

Above is the graph of the ratio of every other term. The graphs of the ratios of terms look very similar. In the second graph, we see the same large jump from the first ratio to the second and then a sharp decrease with the graph leveling off around term 7 at about 2.6.

Now, lets take a look at some other sequences that look like Fibonacci's sequence but begin with other numbers. For example, if we change f(0)=1 and f(1)=3, we form what is termed the Lewis sequence. The first 20 terms of thesequence are:

 term # term in Lewis sequence 1 1 2 3 3 4 4 7 5 11 6 18 7 29 8 47 9 76 10 123 11 199 12 322 13 521 14 843 15 1364 16 2207 17 3571 18 5778 19 9349 20 15127

If we graph the Lewis sequence and compare the graph to the Fibonacci sequence, we have:

With the Fibonacci sequence in purple and the Lewis sequence in blue. We can see that the Lewis sequence has a much steeper slope than the Fibonacci sequence, as n gets larger. However as n is smaller, the two sequences look about the same.

Above is the graphs of the ratios of consecutive terms for both sequences. The Fibonacci sequence is in blue and the Lewis sequence is in purple. It is interesting that for the first few points on the graph, the graphs appear to be reflections of each other, and finally becoming the same function as the graph continues.

The following is the first few terms of the sequence with f(0) = 2 and

f(1) =4. Its graph is compared with the Fibonacci sequence, as well.

 term # 1 2 2 4 3 6 4 10 5 16 6 26 7 42 8 68 9 110 10 178 11 288 12 466 13 754 14 1220 15 1974 16 3194 17 5168 18 8362 19 13530 20 21892

Above is the graph of the ratio of consecutive terms for each of the sequences. The fibonacci sequence is in blue and the new sequence is in purple. Again, we can see that after the first couple of ratios, the two graphs are identical, staying constant near 1.61. It is pretty safe to guess that any recursive sequence, in which you add the two previous terms to get the next, will follow this pattern.

In the following spread sheet, you can enter the first two terms of your fibonacci-like sequence and see that the ratios of the consecutive terms will eventually become 1.61.

Now, lets try a sequence of terms where you subtract the previous two terms to find the next term. The first two terms are 1,000 and 5.

 Sequence ratio of terms 1000 5 0.005 995 199 -990 -0.994974874371859 1985 -2.00505050505051 -2975 -1.49874055415617 4960 -1.6672268907563 -7935 -1.59979838709677 12895 -1.62507876496534 -20830 -1.61535478867778 33725 -1.61905904944791 -54555 -1.61764269829503 88280 -1.61818348455687 -142835 -1.6179768917082 231115 -1.61805579864879 -373950 -1.6180256582221 605065 -1.61803717074475 -979015 -1.6180327733384 1584080 -1.61803445299612 -2563095 -1.61803381142367 4147175 -1.61803405648249 -6710270 -1.61803396287834 10857445 -1.61803399863195

Again, as n gets larger, the sequence tends to -1.618, as we expect it to do.

What I learned: That regardless of the first two terms of a fibonacci sequence, the ratio of consecutive terms will tend to 1.618. If we change the sequence to subtraction of consecutive terms, the ratio of consecutive terms will tend to -1.618.

Things to explore: Why is this the case?

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