# Michelle E. Chung

* EMAT6680 Assignment 12: Fibonnaci Sequence

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

###### a. Construct the ratio of each pair of adjacent terms in the Fibonnaci sequence.      What happens as n increases?      What about the ratio of eery second term? etc.

b. Explore sequences where f(0) and f(1) are some arbitary 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.

a. Fibonnaci Sequence

Fibonnaci Sequence Table
w/its ratios

 n Fibonnaci Sequence f(n)/f(n-1) f(n)/f(n-2) f(n)/f(n-3) f(n)/f(n-4) 1 1 2 1 1 3 2 2 2 4 3 1.5 3 3 5 5 1.666666667 2.5 5 5 6 8 1.6 2.666666667 4 8 7 13 1.625 2.6 4.333333333 6.5 8 21 1.615384615 2.625 4.2 7 9 34 1.619047619 2.615384615 4.25 6.8 10 55 1.617647059 2.619047619 4.230769231 6.875 11 89 1.618181818 2.617647059 4.238095238 6.846153846 12 144 1.617977528 2.618181818 4.235294118 6.857142857 13 233 1.618055556 2.617977528 4.236363636 6.852941176 14 377 1.618025751 2.618055556 4.235955056 6.854545455 15 610 1.618037135 2.618025751 4.236111111 6.853932584 16 987 1.618032787 2.618037135 4.236051502 6.854166667 17 1597 1.618034448 2.618032787 4.236074271 6.854077253 18 2584 1.618033813 2.618034448 4.236065574 6.854111406 19 4181 1.618034056 2.618033813 4.236068896 6.854098361 20 6765 1.618033963 2.618034056 4.236067627 6.854103343 21 10946 1.618033999 2.618033963 4.236068111 6.85410144 22 17711 1.618033985 2.618033999 4.236067926 6.854102167 23 28657 1.61803399 2.618033985 4.236067997 6.85410189 24 46368 1.618033988 2.61803399 4.23606797 6.854101996 25 75025 1.618033989 2.618033988 4.23606798 6.854101955 26 121393 1.618033989 2.618033989 4.236067976 6.854101971 27 196418 1.618033989 2.618033989 4.236067978 6.854101965 28 317811 1.618033989 2.618033989 4.236067977 6.854101967 29 514229 1.618033989 2.618033989 4.236067978 6.854101966 30 832040 1.618033989 2.618033989 4.236067977 6.854101966 31 1346269 1.618033989 2.618033989 4.236067978 6.854101966 32 2178309 1.618033989 2.618033989 4.236067977 6.854101966 33 3524578 1.618033989 2.618033989 4.236067978 6.854101966 34 5702887 1.618033989 2.618033989 4.236067977 6.854101966 35 9227465 1.618033989 2.618033989 4.236067977 6.854101966 Golden Ratio Square of Golden Ratio Cubic of Golden Ratio 4th power of Golden Ratio

Scatter Plot
of the Ratios

 Ratio1 : f(n)/f(n-1) The first ratio, the ratio of each pair of adjecent terms in the Fibonnaci Sequence, is approaching 1.618033989, which is known as Golden Ratio. Ratio2 : f(n)/f(n-2) The second ratio, the ratio of every second term in the Fibonnaci Sequence, is approaching 2.618033989, which is the square of Golden Ratio. Ratio3 : f(n)/f(n-3) The third ratio, the ratio of every third term in the Fibonnaci Sequence, is approaching 4.236067977, which is the cubic of Golden Ratio. Ratio4 : f(n)/f(n-4) The forth ratio, the ratio of every forth term in the Fibonnaci Sequence, is approaching 6.854101966, which is the 4th power of Golden Ratio.

Conclusion

• As n increases, the ratios are bouncing up and down;
however, they are approaching limits of the sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

So, we can predict, if we calculate f(n)/f(n-5), it would be limited to 5th power of Golden Ratio, and so on.

Let's check it.

Table
Scatter Plot
 n f(n)/f(n-5) 1 2 3 4 5 6 8 7 13 8 10.5 9 11.33333333 10 11 11 11.125 12 11.07692308 13 11.0952381 14 11.08823529 15 11.09090909 16 11.08988764 17 11.09027778 18 11.09012876 19 11.09018568 20 11.09016393 21 11.09017224 22 11.09016907 23 11.09017028 24 11.09016982 25 11.09016999 26 11.09016993 27 11.09016995 28 11.09016994 29 11.09016994 30 11.09016994 31 11.09016994 32 11.09016994 33 11.09016994 34 11.09016994 35 11.09016994 5th power of Golden Ratio

From the table, we can assert our conjecture.

• The first terms of the ratios are same as Fibonnaci Sequence.
So, if we take the first term of each ratio and make a sequence, it would be Fibonnaci Sequence.

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b. Lucas Sequence

Lucas Sequence Table
w/its ratios

 n Lucas Sequence f(n)/f(n-1) f(n)/f(n-2) f(n)/f(n-3) f(n)/f(n-4) 1 1 2 3 3 3 4 1.333333333 4 4 7 1.75 2.333333333 7 5 11 1.571428571 2.75 3.666666667 11 6 18 1.636363636 2.571428571 4.5 6 7 29 1.611111111 2.636363636 4.142857143 7.25 8 47 1.620689655 2.611111111 4.272727273 6.714285714 9 76 1.617021277 2.620689655 4.222222222 6.909090909 10 123 1.618421053 2.617021277 4.24137931 6.833333333 11 199 1.617886179 2.618421053 4.234042553 6.862068966 12 322 1.618090452 2.617886179 4.236842105 6.85106383 13 521 1.618012422 2.618090452 4.235772358 6.855263158 14 843 1.618042226 2.618012422 4.236180905 6.853658537 15 1364 1.618030842 2.618042226 4.236024845 6.854271357 16 2207 1.618035191 2.618030842 4.236084453 6.854037267 17 3571 1.61803353 2.618035191 4.236061684 6.854126679 18 5778 1.618034164 2.61803353 4.236070381 6.854092527 19 9349 1.618033922 2.618034164 4.236067059 6.854105572 20 15127 1.618034014 2.618033922 4.236068328 6.854100589 21 24476 1.618033979 2.618034014 4.236067844 6.854102492 22 39603 1.618033992 2.618033979 4.236068029 6.854101765 23 64079 1.618033987 2.618033992 4.236067958 6.854102043 24 103682 1.618033989 2.618033987 4.236067985 6.854101937 25 167761 1.618033989 2.618033989 4.236067975 6.854101977 26 271443 1.618033989 2.618033989 4.236067979 6.854101962 27 439204 1.618033989 2.618033989 4.236067977 6.854101968 28 710647 1.618033989 2.618033989 4.236067978 6.854101966 29 1149851 1.618033989 2.618033989 4.236067977 6.854101966 30 1860498 1.618033989 2.618033989 4.236067978 6.854101966 31 3010349 1.618033989 2.618033989 4.236067977 6.854101966 32 4870847 1.618033989 2.618033989 4.236067978 6.854101966 33 7881196 1.618033989 2.618033989 4.236067977 6.854101966 34 12752043 1.618033989 2.618033989 4.236067978 6.854101966 35 20633239 1.618033989 2.618033989 4.236067977 6.854101966 Golden Ratio Square of Golden Ratio Cubic of Golden Ratio 4th power of Golden Ratio

Scatter Plot
of the Ratios

 Ratio1 : f(n)/f(n-1) The first ratio, the ratio of each pair of adjecent terms in the Lucas Sequence, starts differnt number, 3; however, it is still approaching 1.618033989, which is known as Golden Ratio, just like Fibonnaci Sequence. Ratio2 : f(n)/f(n-2) The second ratio, the ratio of every second term in the Lucas Sequence, starts differnt number, 4; however, it is still approaching 2.618033989, which is square of Golden Ratio, just like Fibonnaci Sequence. Ratio3 : f(n)/f(n-3) The third ratio, the ratio of every third term in the Lucas Sequence, starts differnt number, 7; however, it is still approaching 4.236067977, which is cubic of Golden Ratio, just like Fibonnaci Sequence. Ratio4 : f(n)/f(n-4) The forth ratio, the ratio of every forth term in the Lucas Sequence, starts differnt number, 11; however, it is still approaching 6.854101966, which is 4th power of Golden Ratio, just like Fibonnaci Sequence.

Conclusion

• The Lucas Sequence has different numbers from Fibonnaci Sequence. So, the ratios start from different numbers;
however, they are approaching same limits of the sequence as Fibonnaci Sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

So, we still can predict that if we calculate f(n)/f(n-5), it would be limited to 5th power of Golden Ratio, and so on, just like Fibonnaci Sequence.

• The first terms of the ratios are same as Lucas Sequence.
So, if we take the first term of each ratio and make a sequence, it would be Lucas Sequence.

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c. Other Sequences

When
f(0)=1 and f(1)=10

 n Sequence f(n)/f(n-1) f(n)/f(n-2) f(n)/f(n-3) f(n)/f(n-4) 1 1 2 10 10 3 11 1.1 11 4 21 1.909090909 2.1 21 5 32 1.523809524 2.909090909 3.2 32 6 53 1.65625 2.523809524 4.818181818 5.3 7 85 1.603773585 2.65625 4.047619048 7.727272727 8 138 1.623529412 2.603773585 4.3125 6.571428571 9 223 1.615942029 2.623529412 4.20754717 6.96875 10 361 1.618834081 2.615942029 4.247058824 6.811320755 11 584 1.617728532 2.618834081 4.231884058 6.870588235 12 945 1.618150685 2.617728532 4.237668161 6.847826087 13 1529 1.617989418 2.618150685 4.235457064 6.856502242 14 2474 1.618051014 2.617989418 4.23630137 6.853185596 15 4003 1.618027486 2.618051014 4.235978836 6.854452055 16 6477 1.618036473 2.618027486 4.236102027 6.853968254 17 10480 1.61803304 2.618036473 4.236054972 6.854153041 18 16957 1.618034351 2.61803304 4.236072945 6.854082458 19 27437 1.61803385 2.618034351 4.23606608 6.854109418 20 44394 1.618034042 2.61803385 4.236068702 6.85409912 21 71831 1.618033969 2.618034042 4.236067701 6.854103053 22 116225 1.618033996 2.618033969 4.236068083 6.854101551 23 188056 1.618033986 2.618033996 4.236067937 6.854102125 24 304281 1.61803399 2.618033986 4.236067993 6.854101906 25 492337 1.618033988 2.61803399 4.236067972 6.854101989 26 796618 1.618033989 2.618033988 4.23606798 6.854101957 27 1288955 1.618033989 2.618033989 4.236067977 6.85410197 28 2085573 1.618033989 2.618033989 4.236067978 6.854101965 29 3374528 1.618033989 2.618033989 4.236067977 6.854101967 30 5460101 1.618033989 2.618033989 4.236067978 6.854101966 31 8834629 1.618033989 2.618033989 4.236067977 6.854101966 32 14294730 1.618033989 2.618033989 4.236067978 6.854101966 33 23129359 1.618033989 2.618033989 4.236067977 6.854101966 34 37424089 1.618033989 2.618033989 4.236067978 6.854101966 35 60553448 1.618033989 2.618033989 4.236067977 6.854101966 Golden Ratio Square of Golden Ratio Cubic of Golden Ratio 4th power of Golden Ratio

When
f(0)=10 and f(1)=17

 n Sequence f(n)/f(n-1) f(n)/f(n-2) f(n)/f(n-3) f(n)/f(n-4) 1 10 2 17 1.7 3 27 1.588235294 2.7 4 44 1.62962963 2.588235294 4.4 5 71 1.613636364 2.62962963 4.176470588 7.1 6 115 1.61971831 2.613636364 4.259259259 6.764705882 7 186 1.617391304 2.61971831 4.227272727 6.888888889 8 301 1.61827957 2.617391304 4.23943662 6.840909091 9 487 1.617940199 2.61827957 4.234782609 6.85915493 10 788 1.618069815 2.617940199 4.23655914 6.852173913 11 1275 1.618020305 2.618069815 4.235880399 6.85483871 12 2063 1.618039216 2.618020305 4.23613963 6.853820598 13 3338 1.618031992 2.618039216 4.236040609 6.854209446 14 5401 1.618034751 2.618031992 4.236078431 6.854060914 15 8739 1.618033697 2.618034751 4.236063984 6.854117647 16 14140 1.6180341 2.618033697 4.236069503 6.854095977 17 22879 1.618033946 2.6180341 4.236067395 6.854104254 18 37019 1.618034005 2.618033946 4.2360682 6.854101092 19 59898 1.618033983 2.618034005 4.236067893 6.8541023 20 96917 1.618033991 2.618033983 4.23606801 6.854101839 21 156815 1.618033988 2.618033991 4.236067965 6.854102015 22 253732 1.618033989 2.618033988 4.236067982 6.854101948 23 410547 1.618033989 2.618033989 4.236067976 6.854101973 24 664279 1.618033989 2.618033989 4.236067978 6.854101964 25 1074826 1.618033989 2.618033989 4.236067977 6.854101967 26 1739105 1.618033989 2.618033989 4.236067978 6.854101966 27 2813931 1.618033989 2.618033989 4.236067977 6.854101966 28 4553036 1.618033989 2.618033989 4.236067978 6.854101966 29 7366967 1.618033989 2.618033989 4.236067977 6.854101966 30 11920003 1.618033989 2.618033989 4.236067978 6.854101966 31 19286970 1.618033989 2.618033989 4.236067977 6.854101966 32 31206973 1.618033989 2.618033989 4.236067978 6.854101966 33 50493943 1.618033989 2.618033989 4.236067977 6.854101966 34 81700916 1.618033989 2.618033989 4.236067977 6.854101966 35 132194859 1.618033989 2.618033989 4.236067977 6.854101966 Golden Ratio Square of Golden Ratio Cubic of Golden Ratio 4th power of Golden Ratio

When
f(0)=-7 and f(1)=-3

 n Sequence f(n)/f(n-1) f(n)/f(n-2) f(n)/f(n-3) f(n)/f(n-4) 1 -7 2 -3 0.428571429 3 -10 3.333333333 1.428571429 4 -13 1.3 4.333333333 1.857142857 5 -23 1.769230769 2.3 7.666666667 3.285714286 6 -36 1.565217391 2.769230769 3.6 12 7 -59 1.638888889 2.565217391 4.538461538 5.9 8 -95 1.610169492 2.638888889 4.130434783 7.307692308 9 -154 1.621052632 2.610169492 4.277777778 6.695652174 10 -249 1.616883117 2.621052632 4.220338983 6.916666667 11 -403 1.618473896 2.616883117 4.242105263 6.830508475 12 -652 1.617866005 2.618473896 4.233766234 6.863157895 13 -1055 1.61809816 2.617866005 4.236947791 6.850649351 14 -1707 1.618009479 2.61809816 4.23573201 6.855421687 15 -2762 1.618043351 2.618009479 4.236196319 6.853598015 16 -4469 1.618030413 2.618043351 4.236018957 6.854294479 17 -7231 1.618035355 2.618030413 4.236086702 6.854028436 18 -11700 1.618033467 2.618035355 4.236060825 6.854130053 19 -18931 1.618034188 2.618033467 4.236070709 6.854091238 20 -30631 1.618033913 2.618034188 4.236066934 6.854106064 21 -49562 1.618034018 2.618033913 4.236068376 6.854100401 22 -80193 1.618033978 2.618034018 4.236067825 6.854102564 23 -129755 1.618033993 2.618033978 4.236068036 6.854101738 24 -209948 1.618033987 2.618033993 4.236067955 6.854102053 25 -339703 1.618033989 2.618033987 4.236067986 6.854101933 26 -549651 1.618033989 2.618033989 4.236067974 6.854101979 27 -889354 1.618033989 2.618033989 4.236067979 6.854101961 28 -1439005 1.618033989 2.618033989 4.236067977 6.854101968 29 -2328359 1.618033989 2.618033989 4.236067978 6.854101966 30 -3767364 1.618033989 2.618033989 4.236067977 6.854101967 31 -6095723 1.618033989 2.618033989 4.236067978 6.854101966 32 -9863087 1.618033989 2.618033989 4.236067977 6.854101966 33 -15958810 1.618033989 2.618033989 4.236067978 6.854101966 34 -25821897 1.618033989 2.618033989 4.236067977 6.854101966 35 -41780707 1.618033989 2.618033989 4.236067978 6.854101966 Golden Ratio Square of Golden Ratio Cubic of Golden Ratio 4th power of Golden Ratio

Conclusion

• As we see, no matter what numbers we put into f(0) and f(1), the ratios are approaching the same limits of the sequence as Fibonnaci Sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

>>> Conjecture : The reason why the ratios are approaching the same limits of the sequence as Fibonnaci Sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively, is the ratios come from the recursion, not from the initial numbers.

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