This question is about the Fibonacci sequence 1,1,2,3,5,8,13. . . Choose two successive Fibonacci numbers Fn and Fn+1 for n moderately large (numbers past the end of the part of the sequence shown above) and use the extended Euclidean algorithm to compute x and y such that gcd(Fn, Fn+1) = Fnx + Fn+1y. Then do this again for a larger n (for a larger pair of successive Fibonacci numbers). Describe any patterns that you notice (there will be some points reserved for actually describing some patterns, though most credit will reside in doing two EEA computations). You may be able to conjecture one or two theorems about the Fibonacci sequence after doing your calculations. Extra credit if you can give a formal proof by induction (or using the well-ordering principle) that for any n, gcd(Fn, Fn+1) = 1. There are ways of doing this by talking about the EEA computation, but it can be done directly in fairly simple ways as well (hint: one of the lemmas we used to prove that our algorithm for proving the Euclidean algorithm correct can be used)