Fibonacci's Sequence

by

Hieu Huy Nguyen

Our goal is to investigate the Fibonnaci sequence in order to understand the ratio each pair of adjacent terms in the sequence. The sequence is based off the formula below:

f(n) = f(n-1) + f(n-1)

Our first sequence generated from f(0) = 1, f(1) = 1 was created and displayed below:

Based on the results from the spreadsheet, we can conclude that the ratio of each pair of adjacent terms in the Fibonnaci sequence reaches a limit point where the ratios remain constant as n increases. The same results occur with the ratios of the ever second, third, or fourth term. The ratios seem to end up being a certain constant as n increases. This allows us to assume that for every ratio of each paired terms of the sequence will always approach a certain constant ratio as n increases to infinity.

Next, we want to know if this same behavior holds should the terms for f(0) and f(1) change. Therefore, we choose arbitrary values for f(0) and f(1), and then we re-evaluated the ratio results on the spread sheet again.

For f(0) = 2, f(1) = 2 the results are as follows:

For f(0) = 6, f(1) = 4 the results are as follows:

Based on the results, we can conclude the with variations for f(0) and f(1), the ratios for different paired terms still reach a constant ratio as n increase to infinity. Thus, we can assume that there is only one ratio for each set of paired terms in Fibonnaci’s sequence.

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