## Assignment # 12

Fibonnaci and Phi

Let's look at our sequence for the first 30 terms. As we can see, the ratio of subsequent terms approaches The Golden Ratio closer and closer.

 1 1 1 2 2 3 1.5 5 1.66666666666667 8 1.6 13 1.625 21 1.61538461538462 34 1.61904761904762 55 1.61764705882353 89 1.61818181818182 144 1.61797752808989 233 1.61805555555556 377 1.61802575107296 610 1.61803713527851 987 1.61803278688525 1597 1.61803444782168 2584 1.61803381340013 4181 1.61803405572755 6765 1.61803396316671 10946 1.6180339985218 17711 1.61803398501736 28657 1.6180339901756 46368 1.61803398820532 75025 1.6180339889579 121393 1.61803398867044 196418 1.61803398878024 317811 1.6180339887383 514229 1.61803398875432

What is the Golden Ratio?

 Phi, or the Golden Ratio, is equal to the following constant:

From this, we can see that the successive ratios are getting closer and closer to the true value. Let's take a look at the variance from the actual value as well as the ratio of our values to the true value.

 Fibonnaci f(n-1)/f(n) variance ratio 1 1 1 -0.618033988749895 0.618033988749895 2 2 0.381966011250105 1.23606797749979 3 1.5 -0.118033988749895 0.927050983124842 5 1.66666666666667 0.0486326779167718 1.03005664791649 8 1.6 -0.0180339887498948 0.988854381999832 13 1.625 0.0069660112501051 1.00430523171858 21 1.61538461538462 -0.00264937336527948 0.998362597211369 34 1.61904761904762 0.00101363029772417 1.00062645797602 55 1.61764705882353 -0.000386929926365465 0.999760864154242 89 1.61818181818182 0.000147829431923263 1.00009136361347 144 1.61797752808989 -0.000056460660007307 0.999965105393088 233 1.61805555555556 0.0000215668056606777 1.00001332901893 377 1.61802575107296 -8.23767693347577E-06 0.999994908835667 610 1.61803713527851 3.14652861965747E-06 1.00000194466163 987 1.61803278688525 -1.20186464891425E-06 0.999999257206797 1597 1.61803444782168 4.59071787028975E-07 1.00000028372197 2584 1.61803381340013 -1.75349769593325E-07 0.999999891627882 4181 1.61803405572755 6.69776591966098E-08 1.00000004139447 6765 1.61803396316671 -2.55831884565794E-08 0.99999998418872 10946 1.6180339985218 9.77190839357434E-09 1.00000000603937 17711 1.61803398501736 -3.73253694618825E-09 0.999999997693165 28657 1.6180339901756 1.4257022229458E-09 1.00000000088113 46368 1.61803398820532 -5.44569944693762E-10 0.999999999663437 75025 1.6180339889579 2.08007167046276E-10 1.00000000012856 121393 1.61803398867044 -7.94517784896698E-11 0.999999999950896 196418 1.61803398878024 3.03477243335237E-11 1.00000000001876 317811 1.6180339887383 -1.15918386001113E-11 0.999999999992836 514229 1.61803398875432 4.42756942220512E-12 1.00000000000274

We can also see that our sequence is oscillating around the correct value. Let's examine how this functions behaves:

 By the 8th term, we are pretty much constant ...

Looking at terms other than f(n+1)/f(n) ...

 Fibonnaci f(n+1)/f(n) f(n+2)/f(n) f(n+3)/f(n) f(n+4)/f(n) 1 1 1 2 3 5 2 2 3 5 8 3 1.5 2.5 4 6.5 5 1.66666666666667 2.66666666666667 4.33333333333333 7 8 1.6 2.6 4.2 6.8 13 1.625 2.625 4.25 6.875 21 1.61538461538462 2.61538461538462 4.23076923076923 6.84615384615385 34 1.61904761904762 2.61904761904762 4.23809523809524 6.85714285714286 55 1.61764705882353 2.61764705882353 4.23529411764706 6.85294117647059 89 1.61818181818182 2.61818181818182 4.23636363636364 6.85454545454545 144 1.61797752808989 2.61797752808989 4.23595505617978 6.85393258426966 233 1.61805555555556 2.61805555555556 4.23611111111111 6.85416666666667 377 1.61802575107296 2.61802575107296 4.23605150214592 6.85407725321888 610 1.61803713527851 2.61803713527851 4.23607427055703 6.85411140583554 987 1.61803278688525 2.61803278688525 4.23606557377049 6.85409836065574 1597 1.61803444782168 2.61803444782168 4.23606889564336 6.85410334346505 2584 1.61803381340013 2.61803381340013 4.23606762680025 6.85410144020038 4181 1.61803405572755 2.61803405572755 4.23606811145511 6.85410216718266 6765 1.61803396316671 2.61803396316671 4.23606792633341 6.85410188950012 10946 1.6180339985218 2.6180339985218 4.23606799704361 6.85410199556541 17711 1.61803398501736 2.61803398501736 4.23606797003472 6.85410195505207 28657 1.6180339901756 2.6180339901756 4.23606798035119 6.85410197052679 46368 1.61803398820532 2.61803398820533 4.23606797641065 6.85410196461598 75025 1.6180339889579 2.6180339889579 4.2360679779158 6.85410196687371 121393 1.61803398867044 2.61803398867044 4.23606797734089 6.85410196601133 196418 1.61803398878024 2.61803398878024 4.23606797756049 317811 1.6180339887383 2.6180339887383 514229 1.61803398875432

We can see that taking the f(n+2)/f(n) sequence yields us ... phi + 1. The other terms give us successive powers of phi, i.e.:

This pattern is bound to continue ...

Looking starting points other than f(1) = 1 and f(2) = 1 ...

 Fibonnaci f(n+1)/f(n) 1 3 3 4 1.33333333333333 7 1.75 11 1.57142857142857 18 1.63636363636364 29 1.61111111111111 47 1.62068965517241 76 1.61702127659574 123 1.61842105263158 199 1.61788617886179 322 1.61809045226131 521 1.61801242236025 843 1.61804222648752 1364 1.61803084223013 2207 1.61803519061584 3571 1.6180335296783 5778 1.61803416409969 9349 1.61803392177224 15127 1.61803401433308 24476 1.61803397897799 39603 1.61803399248243 64079 1.61803398732419 103682 1.61803398929446 167761 1.61803398854189 271443 1.61803398882935 439204 1.61803398871955 710647 1.61803398876149 1149851 1.61803398874547
 Fibonnaci f(n+1)/f(n) 2 5 2.5 7 1.4 12 1.71428571428571 19 1.58333333333333 31 1.63157894736842 50 1.61290322580645 81 1.62 131 1.61728395061728 212 1.61832061068702 343 1.61792452830189 555 1.61807580174927 898 1.61801801801802 1453 1.61804008908686 2351 1.6180316586373 3804 1.61803487877499 6155 1.61803364879075 9959 1.61803411860276 16114 1.61803393915052 26073 1.61803400769517 42187 1.61803398151344 68260 1.61803399151397 110447 1.61803398769411 178707 1.61803398915317 289154 1.61803398859586 467861 1.61803398880873 757015 1.61803398872742 1224876 1.61803398875848 1981891 1.61803398874662
 Fibonnaci f(n+1)/f(n) 7 59 8.42857142857143 66 1.11864406779661 125 1.89393939393939 191 1.528 316 1.6544502617801 507 1.60443037974684 823 1.6232741617357 1330 1.61603888213852 2153 1.6187969924812 3483 1.6177426846261 5636 1.61814527706001 9119 1.6179914833215 14755 1.61805022480535 23874 1.61802778719078 38629 1.61803635754377 62503 1.61803308395247 101132 1.61803433435195 163635 1.61803385674168 264767 1.61803403917255 428402 1.61803396949016 693169 1.61803399610646 1121571 1.61803398593994 1814740 1.6180339898232 2936311 1.61803398833993 4751051 1.61803398890649 7687362 1.61803398869008 12438413 1.61803398877274 20125775 1.61803398874117

It looks like it doesn't matter which values we choose as our starting point - the sequence is bound to converge to phi sooner or later. Let's look at why that's the case ...

 a b a+b a + 2b 2a + 3b 3a + 5b 5a + 8b 8a + 13b 13a + 21b 21a + 34b 34a + 55b etc É

The coefficients of both the 'a' and 'b' terms are consecutive numbers from the Fibonnaci sequence. Dividing f(n+1)/f(n) will start approaching phi. Why is that the case?

Here goes our proof ...

The positive value will give us Phi.