Fibonnaci Sequence

Rozina Essani

For this investigation I looked at the Fibonnaci sequence and applied it to Excel spreadsheets. I have investigated the sequence for when n=0,1,2,…,47. View the Excel workdsheets here.

Fibonnaci Sequence

 n f(n)=f(n-1)+f(n+2) f(n)/f(n+1) f(n)/f(n+2) f(n)/f(n+3) f(n)/f(n+4) 0 1 1 1 1 2 2 0.5 0.5 3 3 0.666666667 0.333333333 0.333333333 4 5 0.6 0.4 0.2 0.2 5 8 0.625 0.375 0.25 0.125

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 42 433494437 0.618034 0.381966 0.236068 0.145898 43 701408733 0.618034 0.381966 0.236068 0.145898 44 1134903170 0.618034 0.381966 0.236068 0.145898 45 1836311903 0.618034 0.381966 0.236068 0.145898 46 2971215073 0.618034 0.381966 0.236068 0.145898 47 4807526976 0.618034 0.381966 0.236068 0.145898

The Fibonnaci sequence is defined as f(n) = f(n-1) + f(n-2) where f(0)=1 and f(1)=1. The sequence adds up to a number in billions when n reaches 47.

Let us first investigate the ratio of each pair of adjacent terms in the Fibonnaci sequence. We see that the ratio, f(n)/f(n+1) has a limit which is 0.618033989. Now let us look at the ratio of every second term in the Fibonnaci sequence. The first ratio of f(n)/f(n+2) is the second ratio of f(n)/f(n+1). This ratio’s limiting number is 0.381966011. Now let us look at the ratio of every third term and every fourth term. Again the first entries for these ratios are the second entries of the previous ratios respectively. The limit for f(n)/f(n+3) and f(n)/f(n+4) is 0.23606798 and 0.14589503 respectively.

When changing f(1)=1 to f(1)=3, the Lucas sequence, our ratios have the same limits even though they do not began the same way as in the Fibonnaci sequence.

Lucas Sequence

 n f(n)=f(n-1)+f(n+2) f(n)/f(n+1) f(n)/f(n+2) 0 1 1 3 0.333333333 2 4 0.75 0.25 3 7 0.571428571 0.428571429 4 11 0.636363636 0.363636364 5 18 0.611111111 0.388888889

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 42 969323029 0.618034 0.381966 43 1568397607 0.618034 0.381966 44 2537720636 0.618034 0.381966 45 4106118243 0.618034 0.381966 46 6643838879 0.618034 0.381966 47 10749957122 0.618034 0.381966

Now if we take the ratios of each pair of adjacent limits then we get a value of 1.61803399. This in fact is the Golden Ratio.