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.

 0.384106 0.384615 0.238095238095238 0.147058823529412 0.381148 0.380952 0.235294117647059 0.145454545454545 0.382278 0.382353 0.236363636363636 0.146067415730337 0.381847 0.381818 0.235955056179775 0.145833333333333 0.382012 0.382022 0.236111111111111 0.145922746781116 0.381949 0.381944 0.236051502145923 0.145888594164456 0.381973 0.381974 0.236074270557029 0.145901639344262 0.381963 0.381963 0.236065573770492 0.145896656534954 0.381967 0.381967 0.236068895643364 0.145898559799624 0.381966 0.381966 0.23606762680025 0.145897832817337 0.381966 0.381966 0.236068111455108 0.14589811049988 0.381966 0.381966 0.236067926333413 0.14589800443459 0.381966 0.381966 0.236067997043607 0.145898044947926 0.381966 0.381966 0.236067970034716 0.145898029473209 0.381966 0.381966 0.236067980351194 0.145898035384025 0.381966 0.381966 0.23606797641065 0.145898033126294 0.381966 0.381966 0.236067977915804 0.14589803398867 0.381966 0.381966 0.236067977340886 0.145898033659272 0.381966 0.381966 0.236067977560485 0.145898033785091 0.381966 0.381966 0.236067977476606 0.145898033737032 0.381966 0.381966 0.236067977508645 0.145898033755389 0.381966 0.381966 0.236067977496407 0.145898033748377 0.381966 0.381966 0.236067977501082 0.145898033751056 0.381966 0.381966 0.236067977499296 0.145898033750033 0.381966 0.381966 0.236067977499978 0.145898033750423 0.381966 0.381966 0.236067977499718 0.145898033750274 0.381966 0.381966 0.236067977499817 0.145898033750331 0.381966 0.381966 0.236067977499779 0.145898033750309 0.381966 0.381966 0.236067977499794 0.145898033750318 0.381966 0.381966 0.236067977499788 0.145898033750315 0.381966 0.381966 0.23606797749979 0.145898033750316 0.381966 0.381966 0.236067977499789 0.381966 0.381966
.

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