EMAT 6680 Assignment 12

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.

 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 infinity

and we are left with

Hence, the ratio is the golden ratio.

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