By

Brooke Norman

Program such as Excel are excellent spreadsheet programs that can be used to generate tables of numbers.

In this assignment, we are going to use Excel to generate a Fibonnaci sequence.

I used Microsoft Excel to generate a Fibonnaci sequence from n=1 to n=34.  You can see the sequence in the table below.  How did I do this?  First, I listed a number 1 in block A2.  In block A3, I entered a 1 also.  In block A4, I entered the formula =A2+A3.  This gives block A4 a value of 2.   I then highlighted the blocks in column A from A4 down to A34. I then used the “fill down” command to copy the formula into each of these blocks.

I can check this to make sure it is correct.  Let’s choose block A23 for example.  We will add block A21+A22 and see if it gives us A23.  [6765+10946=17711].  If we look at the number in A23, the value is 17711.  The values are correct!

I then found the ratio of each pair of adjacent terms in the Fibonnaci sequence.  This was done by entering a 1 in block B2.  In block B3, I entered the equation of =A3/A2.  I used the same technique of highlighting block B3 and dragging the highlight all the way down to B34.  Take a look at the different ratios.  It appears that one increases and the next decreases and continues in that pattern until they begin to reach common number.  This is called a limit. They seem to approach a limit of 1.618033989 as n becomes larger or approaches infinity.

 Fibonnaci Seq Ratio 1 1 1 1 2 2 3 1.5 5 1.666666667 8 1.6 13 1.625 21 1.615384615 34 1.619047619 55 1.617647059 89 1.618181818 144 1.617977528 233 1.618055556 377 1.618025751 610 1.618037135 987 1.618032787 1597 1.618034448 2584 1.618033813 4181 1.618034056 6765 1.618033963 10946 1.618033999 17711 1.618033985 28657 1.61803399 46368 1.618033988 75025 1.618033989 121393 1.618033989 196418 1.618033989 317811 1.618033989 514229 1.618033989 832040 1.618033989 1346269 1.618033989 2178309 1.618033989 3524578 1.618033989

Next, I chose some arbitrary integers for f(0) and f(1), other than 1. I decided to make f(0)=4 and f(1)=5. Notice that in the first column, the values are very different from the values of the Fibonnaci sequence in the first table. But take notice that the ratio of the adjacent terms in the second column still approach the same limit of l.618, just like the Fibonnaci sequence.

 Fibonnaci Seq Ratio Seq. 2 Ratio 2 1 1 4 1 1 5 1.25 2 2 9 1.8 3 1.5 14 1.555555556 5 1.666666667 23 1.642857143 8 1.6 37 1.608695652 13 1.625 60 1.621621622 21 1.615384615 97 1.616666667 34 1.619047619 157 1.618556701 55 1.617647059 254 1.617834395 89 1.618181818 411 1.618110236 144 1.617977528 665 1.618004866 233 1.618055556 1076 1.618045113 377 1.618025751 1741 1.61802974 610 1.618037135 2817 1.618035612 987 1.618032787 4558 1.618033369 1597 1.618034448 7375 1.618034226 2584 1.618033813 11933 1.618033898 4181 1.618034056 19308 1.618034023 6765 1.618033963 31241 1.618033976 10946 1.618033999 50549 1.618033994 17711 1.618033985 81790 1.618033987 28657 1.61803399 132339 1.618033989 46368 1.618033988 214129 1.618033988 75025 1.618033989 346468 1.618033989 121393 1.618033989 560597 1.618033989 196418 1.618033989 907065 1.618033989 317811 1.618033989 1467662 1.618033989 514229 1.618033989 2374727 1.618033989 832040 1.618033989 3842389 1.618033989 1346269 1.618033989 6217116 1.618033989 2178309 1.618033989 10059505 1.618033989 3524578 1.618033989 16276621 1.618033989

For the last part of this assignment, I explored a sequence where f(0)=1 and f(1)=3.  This is called a Lucas Sequence.  Take notice that as n gets larger or approaches infinity, the ratio reaches the same limit as the Fibonnaci sequence.

 Fibonnaci Seq Fib. Ratio Lucas Seq Luc. Ratio 1 1 1 1 1 3 3 2 2 4 1.333333333 3 1.5 7 1.75 5 1.666666667 11 1.571428571 8 1.6 18 1.636363636 13 1.625 29 1.611111111 21 1.615384615 47 1.620689655 34 1.619047619 76 1.617021277 55 1.617647059 123 1.618421053 89 1.618181818 199 1.617886179 144 1.617977528 322 1.618090452 233 1.618055556 521 1.618012422 377 1.618025751 843 1.618042226 610 1.618037135 1364 1.618030842 987 1.618032787 2207 1.618035191 1597 1.618034448 3571 1.61803353 2584 1.618033813 5778 1.618034164 4181 1.618034056 9349 1.618033922 6765 1.618033963 15127 1.618034014 10946 1.618033999 24476 1.618033979 17711 1.618033985 39603 1.618033992 28657 1.61803399 64079 1.618033987 46368 1.618033988 103682 1.618033989 75025 1.618033989 167761 1.618033989 121393 1.618033989 271443 1.618033989 196418 1.618033989 439204 1.618033989 317811 1.618033989 710647 1.618033989 514229 1.618033989 1149851 1.618033989 832040 1.618033989 1860498 1.618033989 1346269 1.618033989 3010349 1.618033989 2178309 1.618033989 4870847 1.618033989 3524578 1.618033989 7881196 1.618033989

In closing, we can say that the limit of the ratio for the adjacent terms of the Fibonnaci sequence and the Lucas sequence both approach 1.618, as well as, other similar sequences, as n goes to infinity.