Assignment #12

For EMAT 6680
Authored By

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.