Paula Whitmire
Assignment 12
Fibonacci Numbers
Column A and Column B
| 0 |
|
| 1 |
0 |
| 1 |
1 |
| 2 |
0.5 |
| 3 |
0.666666666666667 |
| 5 |
0.6 |
| 8 |
0.625 |
| 13 |
0.615384615384615 |
| 21 |
0.619047619047619 |
| 34 |
0.617647058823529 |
| 55 |
0.618181818181818 |
| 89 |
0.617977528089888 |
| 144 |
0.618055555555556 |
| 233 |
0.618025751072961 |
| 377 |
0.618037135278515 |
| 610 |
0.618032786885246 |
| 987 |
0.618034447821682 |
| 1597 |
0.618033813400125 |
| 2584 |
0.618034055727554 |
| 4181 |
0.618033963166707 |
| 6765 |
0.618033998521803 |
| 10946 |
0.618033985017358 |
| 17711 |
0.618033990175597 |
| 28657 |
0.618033988205325 |
| 46368 |
0.618033988957902 |
| 75025 |
0.618033988670443 |
| 121393 |
0.618033988780243 |
| 196418 |
0.618033988738303 |
| 317811 |
0.618033988754323 |
| 514229 |
0.618033988748204 |
| 832040 |
0.618033988750541 |
| 1346269 |
0.618033988749648 |
| 2178309 |
0.618033988749989 |
| 3524578 |
0.618033988749859 |
| 5702887 |
0.618033988749909 |
| 9227465 |
0.61803398874989 |
| 14930352 |
0.618033988749897 |
| 24157817 |
0.618033988749894 |
| 39088169 |
0.618033988749895 |
| 63245986 |
0.618033988749895 |
The first 40 Fibonacci Numbers are in the first column
(a) above. The fibonacci sequence is the recursive A(n) = A (n-2)
+ A ( n-1). In column 2 (b) above, the terms are the ratio of
each pair of adjacent terms in column 1. For example, b2=a1/a2
or b7=a6/a7. The ratio converges to .61803 which is the golden
ratio.
Now we will pick the first two numbers for Column C
as 1 and 3. We will use the same recursive formula for Columnc
C. Column D will again be the ratio of the adjacent pair.
Again the ratio approaches the golden ratio of .61803.
Column C and Column D
| 1 |
|
| 3 |
0.333333333333333 |
| 4 |
0.75 |
| 7 |
0.571428571428571 |
| 11 |
0.636363636363636 |
| 18 |
0.611111111111111 |
| 29 |
0.620689655172414 |
| 47 |
0.617021276595745 |
| 76 |
0.618421052631579 |
| 123 |
0.617886178861789 |
| 199 |
0.618090452261307 |
| 322 |
0.618012422360248 |
| 521 |
0.618042226487524 |
| 843 |
0.61803084223013 |
| 1364 |
0.618035190615836 |
| 2207 |
0.618033529678296 |
| 3571 |
0.618034164099692 |
| 5778 |
0.618033921772239 |
| 9349 |
0.618034014333084 |
| 15127 |
0.618033978977986 |
| 24476 |
0.618033992482432 |
| 39603 |
0.618033987324193 |
| 64079 |
0.618033989294465 |
| 103682 |
0.618033988541888 |
| 167761 |
0.618033988829346 |
| 271443 |
0.618033988719547 |
| 439204 |
0.618033988761487 |
| 710647 |
0.618033988745467 |
| 1149851 |
0.618033988751586 |
| 1860498 |
0.618033988749249 |
| 3010349 |
0.618033988750142 |
| 4870847 |
0.618033988749801 |
| 7881196 |
0.618033988749931 |
| 12752043 |
0.618033988749881 |
| 20633239 |
0.6180339887499 |
| 33385282 |
0.618033988749893 |
| 54018521 |
0.618033988749896 |
| 87403803 |
0.618033988749895 |
| 141422324 |
0.618033988749895 |
| 228826127 |
0.618033988749895 |
The process was repeated in Excel using constants of
2,5 and 2,14 and -5,12 and 2,22 as the first 2 elements of the
column. Finding the ratio of the adjacent pair gave the golden
ratio value of .61803 every time.
Columns M N O P
are below.
| 0.384105960264901 |
0.384615384615385 |
0.238095238095238 |
0.147058823529412 |
| 0.381147540983607 |
0.380952380952381 |
0.235294117647059 |
0.145454545454545 |
| 0.382278481012658 |
0.382352941176471 |
0.236363636363636 |
0.146067415730337 |
| 0.381846635367762 |
0.381818181818182 |
0.235955056179775 |
0.145833333333333 |
| 0.382011605415861 |
0.382022471910112 |
0.236111111111111 |
0.145922746781116 |
| 0.381948595337717 |
0.381944444444444 |
0.236051502145923 |
0.145888594164456 |
| 0.381972663465091 |
0.381974248927039 |
0.236074270557029 |
0.145901639344262 |
| 0.381963470319635 |
0.381962864721485 |
0.236065573770492 |
0.145896656534954 |
| 0.381966981797658 |
0.381967213114754 |
0.236068895643364 |
0.145898559799624 |
| 0.381965640533705 |
0.381965552178318 |
0.23606762680025 |
0.145897832817337 |
| 0.381966152851137 |
0.381966186599875 |
0.236068111455108 |
0.14589811049988 |
| 0.381965957163319 |
0.381965944272446 |
0.236067926333413 |
0.14589800443459 |
| 0.381966031909418 |
0.381966036833293 |
0.236067997043607 |
0.145898044947926 |
| 0.38196600335895 |
0.381966001478197 |
0.236067970034716 |
0.145898029473209 |
| 0.381966014264258 |
0.381966014982642 |
0.236067980351194 |
0.145898035384025 |
| 0.381966010098801 |
0.381966009824403 |
0.23606797641065 |
0.145898033126294 |
| 0.381966011689864 |
0.381966011794675 |
0.236067977915804 |
0.14589803398867 |
| 0.381966011082132 |
0.381966011042098 |
0.236067977340886 |
0.145898033659272 |
| 0.381966011314265 |
0.381966011329557 |
0.236067977560485 |
0.145898033785091 |
| 0.381966011225598 |
0.381966011219757 |
0.236067977476606 |
0.145898033737032 |
| 0.381966011259466 |
0.381966011261697 |
0.236067977508645 |
0.145898033755389 |
| 0.38196601124653 |
0.381966011245677 |
0.236067977496407 |
0.145898033748377 |
| 0.381966011251471 |
0.381966011251796 |
0.236067977501082 |
0.145898033751056 |
| 0.381966011249584 |
0.381966011249459 |
0.236067977499296 |
0.145898033750033 |
| 0.381966011250304 |
0.381966011250352 |
0.236067977499978 |
0.145898033750423 |
| 0.381966011250029 |
0.381966011250011 |
0.236067977499718 |
0.145898033750274 |
| 0.381966011250134 |
0.381966011250141 |
0.236067977499817 |
0.145898033750331 |
| 0.381966011250094 |
0.381966011250091 |
0.236067977499779 |
0.145898033750309 |
| 0.381966011250109 |
0.38196601125011 |
0.236067977499794 |
0.145898033750318 |
| 0.381966011250104 |
0.381966011250103 |
0.236067977499788 |
0.145898033750315 |
| 0.381966011250106 |
0.381966011250106 |
0.23606797749979 |
0.145898033750316 |
| 0.381966011250105 |
0.381966011250105 |
0.236067977499789 |
|
| 0.381966011250105 |
0.381966011250105 |
|
|
.
Further examination reveals the following:
Column M is term K1/K3 and yields .38196. Then if you
calculate .38196/.61803 you obtain
.618028 which is very close to the golden ratio again.
Next, we try Column N and calculate the term A1/A3
and get .38196.
Then in Column O we evaluate A1/A4 and get .23606. Take this value
and divide by the golden ratio: .23606/.61803 = .38196 !!!!!!!
Trying Column P with A1/A5 , we get a converging value
of .14705.
This value divided by the previous colum value of .23606
yields .622935, very close to the golden ratio. If more place
values were used, then a more accurate value could be found.
RETURN