Write-Up #12:
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 | 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.