In this exploration, we will be looking at the Fibonnaci Sequence. The Fibonacci number, or sequences are defined by the following recurrence relation:
Each number is the sum of the two preceding numbers. These numbers are named after Leonardo of Pisa.
So if we look
at the table below, we will see that the first column is set f(0) = 1 and f(1) = 1 and the formula
used was 
. You will notice that each column
converges to a value.
| 1 | ||||
| 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 |
We can see that as n increases, the first column's ratio converges to the golden ratio (1.618), the second column converges to 2.618, the third column converges to 4.23 and the last column converges to 6.854.
Now, we want to see sequences where f(0) and f(1) are arbitary integers other than 1. The sequence is a Lucas sequence for f(0) = 1 and f(1) = 3 and all such sequences have the same limit of ratio of successive terms.
| 1 | ||||
| 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 |
| 6 | ||||
| 5 | 0.833333333333333 | |||
| 11 | 2.2 | 1.83333333333333 | ||
| 16 | 1.45454545454545 | 3.2 | 2.66666666666667 | |
| 27 | 1.6875 | 2.45454545454545 | 5.4 | 4.5 |
| 43 | 1.59259259259259 | 2.6875 | 3.90909090909091 | 8.6 |
| 70 | 1.62790697674419 | 2.59259259259259 | 4.375 | 6.36363636363636 |
| 113 | 1.61428571428571 | 2.62790697674419 | 4.18518518518519 | 7.0625 |
| 183 | 1.61946902654867 | 2.61428571428571 | 4.25581395348837 | 6.77777777777778 |
| 296 | 1.61748633879781 | 2.61946902654867 | 4.22857142857143 | 6.88372093023256 |
| 479 | 1.61824324324324 | 2.61748633879781 | 4.23893805309735 | 6.84285714285714 |
| 775 | 1.61795407098121 | 2.61824324324324 | 4.23497267759563 | 6.85840707964602 |
| 1254 | 1.61806451612903 | 2.61795407098121 | 4.23648648648649 | 6.85245901639344 |
| 2029 | 1.61802232854864 | 2.61806451612903 | 4.23590814196242 | 6.85472972972973 |
| 3283 | 1.61803844258255 | 2.61802232854864 | 4.23612903225806 | 6.85386221294363 |
| 5312 | 1.61803228754188 | 2.61803844258255 | 4.23604465709729 | 6.8541935483871 |
| 8595 | 1.61803463855422 | 2.61803228754188 | 4.23607688516511 | 6.85406698564593 |
Ok, so we know
that as n increases, 
converges
to 1.618, 
converges to 2.61, 
converges
to 4.23, 
converges to 6.85, and 
converges
to 11.09 and so on.
Nice, OUI?