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Let *F _{n}* be the

*n*

^{th}Fibonacci number where

*F*

_{1}=

*F*

_{2}=1, and

*F*

_{n}_{+1}=

*F*+

_{n}*F*

_{n}_{-1}.

We should prove that

i.e. the golden ratio. We have that

or, if

then

and then you solve this for *x*. You will get a quadratic that has a negative and a positive solution, but since all terms are positive, the positive root must be the right one, and as you can verify it indeed the golden ratio.

Interestingly enough we never used the starting numbers in proving this. If one is negative we might get a sequence of negative numbers, but the result will still hold true. The only exception is if both the starting numbers are 0.

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