I generated the Fibonnaci sequence using EXCEL spreadsheet. In the first column, I set f(0)=1 and f(1)=1. The rest of the sequence is generated using the formula f(n) = f(n-1) + f(n-2). In columns 2, I construct the ratio of each pair of adjacent terms in Fibonnaci. As n increases, the ratio converges to 1.618. The ratio of every second , third and fourth term converges to 2.618 , 4.23, 6.854 respectively as n increases.
Click Here to see the results in EXCEL.
| 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 |
Next, I explore sequences where f(0) and f(1) are arbitary integers other than 1. For f(0)=1 and f(1)=3, the sequence is a Lucas sequence and all such sequence 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 |
The following obervations are made as n tends to larger values:
converges to 1.618
converges to 2.61
converges to 4.23
converges to 6.85
converges to 11.09
When
n= 2, f(2) = f(1) + f(0)
n= 3, f(3) = 2f(1) + f(0)
n= 4, f(4) = 3f(1) + 2f(0)
n= 5, f(5) = 5f(1) + 3f(0)
n= 6, f(6) = 8f(1) + 5f(0)
n= 7, f(7) = 13f(1) + 8f(0)
n= 8, f(8) = 21f(1) + 13f(0)
We want to prove that
converges to 1.618 as n increases
Proof by induction:
n=2
< 2
n=3
< 2
General case:
< 2
We want to prove that
converges to 2.61 as n increases
Proof by induction:
n=1
< 3
n=2
< 3
General case:
< 3
Similarly, we can prove that
< 5
I would like to explore a little more on:
converges to 1.618
This ratio approaches a limit value of 1.618. this is actually
the golden ratio 
.
Recently I happen to come across a page from Alfred S. Posamentier, that attempts to connect Fibonacci sequence and geometery. It states the following result:
If
, 
then the above ratio becomes
That is,
I would like to analyze this expression and show that it tends to
as n tends to large values.Using limit thereom,
tends to zero as n tends to infinity
tends to zero as n tends to infinityand we are left with
Hence, the ratio is the golden ratio.