Write-Up #12:

Mathematics Problems and Explorations with Spreadsheets

by: BJ Jackson

Fibonnaci and Lucas Sequences

Fibonnaci and Lucas sequences have the same general form. Given the first two numbers in a sequence, the next term in the sequence is equal to the sum of the two previous terms. A spreadsheet is useful because the values in the sequences become extremely large in a relatively short amount of time. Spreadsheets also allow one to manipulate the terms quickly and easily.

One interesting aspect of these sequences is the ratios of consecutive terms. The ratios of consecutive terms will eventually converge to the golden ratio approximately 1.61803399. This happens in both types of sequences no matter what two values are chosen to start a Lucas Sequence. The only thing that varies is how quickly the sequence converges to the golden ratio. Below is an example of the Fibonnacci Sequence.

Terms Ratio of Consecutive Terms
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
832040 1.6180339887482

This chart has the first 30 terms of the Fibonnaci sequence. There are a coupl of interesting things to notice in the chart.

First, as claimed above, the number do converge to the golden ratio when rounded to seven decimal places. Second, the ratios alternate. The first ratio is less than the golden ratio while the second ratio is greater than the golden ratio. The third ratio is again less than the golden ratio but larger than the first ratio while the fourth ratio is again greater than the golden ratio but less than the second ratio. This pattern continues until the two sides converge at the golden ratio.

Click here for a link to an Excel spreadsheet that has the Fibonnaci sequence and some examples of Lucas sequences. If you would like to create your own Lucas sequence just change the first two numbers in any of the sequences and the spreadsheet will create a sequence with the first thirty terms of the sequence.

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