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.
