For some reason my friend and I have both been obsessing lately about the Fibonacci sequence and it's relationship with the golden ratio phi. Last fall I played around with the numbers, trying to find patterns and relationships. I came up with two: F[12] = 144 -> a case where the value of F[n] is n^2. Somewhat interesting. I remembered finding another relationship between numbers of the Fibonacci Sequence which provided phi other than the ratio of sequential Fibonacci numbers. I thought I had found something unique so I went to the library today to do some research to see if my formulation was unique. Alas, it wasn't, so I feel comfortable posting here. Here is what I came up with:
My formulation:
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lim(n -> inf): (F[n+a]/F[n])^(1/a) = phi for any initial value of n.
This means that if n is very large, then you can approximate phi from any two numbers in the sequence. 'a' is difference in location in the sequence of the two Fibonacci numbers. By taking the 'a'th root of the ratio of the two numbers, you get phi.
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The above is a more general case of the well known solution (where a=1):
lim(n -> inf): (F[n+1]/F[n]) = phi for any initial value of n
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