Assignment #12

For EMAT 6680
Authored By

Kevin Adams


The Fibonnaci Sequence

Recall that the Fibonnaci Sequence starts with the first two numbers being one. Each subsequent number is the sum of the two previous numbers. This sequence can be easily generated using a spreadsheet such as Microsoft Excel. Another interesting result to consider is the ratio of pairs of numbers in the sequence. In the spreadsheet below we can look at the ratio of every other number, every second number, every third number, and every fourth number. It is interesting that in each case the ratio seems to be converging on a specific number.

1 Ratio of Every Other Term Ratio of Every Second Term Ratio of Every Third Term Ratio of Every Fourth Term
1 1
2 2 2
3 1.5 3 3
5 1.66666666666667 2.5 5 5
8 1.6 2.66666666666667 4 8
13 1.625 2.6 4.33333333333333 6.5
21 1.61538461538462 2.625 4.2 7
34 1.61904761904762 2.61538461538462 4.25 6.8
55 1.61764705882353 2.61904761904762 4.23076923076923 6.875
89 1.61818181818182 2.61764705882353 4.23809523809524 6.84615384615385
144 1.61797752808989 2.61818181818182 4.23529411764706 6.85714285714286
233 1.61805555555556 2.61797752808989 4.23636363636364 6.85294117647059
377 1.61802575107296 2.61805555555556 4.23595505617978 6.85454545454545
610 1.61803713527851 2.61802575107296 4.23611111111111 6.85393258426966
987 1.61803278688525 2.61803713527851 4.23605150214592 6.85416666666667
1597 1.61803444782168 2.61803278688525 4.23607427055703 6.85407725321888
2584 1.61803381340013 2.61803444782168 4.23606557377049 6.85411140583554
4181 1.61803405572755 2.61803381340013 4.23606889564336 6.85409836065574
6765 1.61803396316671 2.61803405572755 4.23606762680025 6.85410334346505
10946 1.6180339985218 2.61803396316671 4.23606811145511 6.85410144020038
17711 1.61803398501736 2.6180339985218 4.23606792633341 6.85410216718266
28657 1.6180339901756 2.61803398501736 4.23606799704361 6.85410188950012
46368 1.61803398820532 2.6180339901756 4.23606797003472 6.85410199556541
75025 1.6180339889579 2.61803398820533 4.23606798035119 6.85410195505207
121393 1.61803398867044 2.6180339889579 4.23606797641065 6.85410197052679
196418 1.61803398878024 2.61803398867044 4.2360679779158 6.85410196461598
317811 1.6180339887383 2.61803398878024 4.23606797734089 6.85410196687371
514229 1.61803398875432 2.6180339887383 4.23606797756049 6.85410196601133
832040 1.6180339887482 2.61803398875432 4.23606797747661 6.85410196634073
1346269 1.61803398875054 2.6180339887482 4.23606797750864 6.85410196621491
2178309 1.61803398874965 2.61803398875054 4.23606797749641 6.85410196626297
3524578 1.61803398874999 2.61803398874965 4.23606797750108 6.85410196624461
5702887 1.61803398874986 2.61803398874999 4.2360679774993 6.85410196625162
9227465 1.61803398874991 2.61803398874986 4.23606797749998 6.85410196624894
14930352 1.61803398874989 2.61803398874991 4.23606797749972 6.85410196624997
24157817 1.6180339887499 2.61803398874989 4.23606797749982 6.85410196624958
39088169 1.61803398874989 2.6180339887499 4.23606797749978 6.85410196624973
63245986 1.6180339887499 2.61803398874989 4.23606797749979 6.85410196624967
102334155 1.61803398874989 2.6180339887499 4.23606797749979 6.85410196624969
Ratio Converges To: 1.618033989 2.618033989 4.236067977 6.854101966


The Lucas Sequence

If we change the second number of the Fibonnaci Sequence from a 1 to a 3 but still make each subsequent term the sum of the previous two numbers then this will generate the Lucas Sequence. Perhaps the most interesting result is that the ratio of the pairs of numbers remains the same even when we change the second number.

1 Ratio of Every Other Term Ratio of Every Second Term Ratio of Every Third Term Ratio of Every Fourth Term
3 3
4 1.33333333333333 4
7 1.75 2.33333333333333 7
11 1.57142857142857 2.75 3.66666666666667 11
18 1.63636363636364 2.57142857142857 4.5 6
29 1.61111111111111 2.63636363636364 4.14285714285714 7.25
47 1.62068965517241 2.61111111111111 4.27272727272727 6.71428571428571
76 1.61702127659574 2.62068965517241 4.22222222222222 6.90909090909091
123 1.61842105263158 2.61702127659574 4.24137931034483 6.83333333333333
199 1.61788617886179 2.61842105263158 4.23404255319149 6.86206896551724
322 1.61809045226131 2.61788617886179 4.23684210526316 6.85106382978723
521 1.61801242236025 2.61809045226131 4.23577235772358 6.85526315789474
843 1.61804222648752 2.61801242236025 4.23618090452261 6.85365853658537
1364 1.61803084223013 2.61804222648752 4.2360248447205 6.85427135678392
2207 1.61803519061584 2.61803084223013 4.23608445297505 6.85403726708075
3571 1.6180335296783 2.61803519061584 4.23606168446026 6.85412667946257
5778 1.61803416409969 2.6180335296783 4.23607038123167 6.85409252669039
9349 1.61803392177224 2.61803416409969 4.23606705935659 6.85410557184751
15127 1.61803401433308 2.61803392177224 4.23606832819938 6.85410058903489
24476 1.61803397897799 2.61803401433308 4.23606784354448 6.85410249229908
39603 1.61803399248243 2.61803397897799 4.23606802866617 6.85410176531672
64079 1.61803398732419 2.61803399248243 4.23606795795597 6.85410204299925
103682 1.61803398929446 2.61803398732419 4.23606798496486 6.85410193693396
167761 1.61803398854189 2.61803398929446 4.23606797464839 6.8541019774473
271443 1.61803398882935 2.61803398854189 4.23606797858893 6.85410196197258
439204 1.61803398871955 2.61803398882935 4.23606797708378 6.85410196788339
710647 1.61803398876149 2.61803398871955 4.23606797765869 6.85410196562566
1149851 1.61803398874547 2.61803398876149 4.23606797743909 6.85410196648804
1860498 1.61803398875159 2.61803398874547 4.23606797752297 6.85410196615864
3010349 1.61803398874925 2.61803398875159 4.23606797749093 6.85410196628446
4870847 1.61803398875014 2.61803398874925 4.23606797750317 6.8541019662364
7881196 1.6180339887498 2.61803398875014 4.2360679774985 6.85410196625476
12752043 1.61803398874993 2.6180339887498 4.23606797750028 6.85410196624775
20633239 1.61803398874988 2.61803398874993 4.2360679774996 6.85410196625042
33385282 1.6180339887499 2.61803398874988 4.23606797749986 6.8541019662494
54018521 1.61803398874989 2.6180339887499 4.23606797749976 6.85410196624979
87403803 1.6180339887499 2.61803398874989 4.2360679774998 6.85410196624964
141422324 1.61803398874989 2.6180339887499 4.23606797749979 6.8541019662497
228826127 1.61803398874989 2.61803398874989 4.23606797749979 6.85410196624968
Ratio Converges To: 1.618033989 2.618033989 4.236067977 6.854101966


What if...

At this point you may be wondering if the ratios are going to remain the same when we start with any two numbers. If you wish to test your chosen two numbers then click here to open up the Excel file used in constructing this web page. Another aspect you may have noticed is that the ratio given in the second column is converging on what is known as the Golden Ratio. If you are interested in more information on the Golden Ratio then click here.

 


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