(50g) Nth Fibonacci Number

03022015, 02:11 AM
Post: #19




RE: (50g) Nth Fibonacci Number
(02272015 01:48 PM)rprosperi Wrote: So it appears an ansatz is a postulated theorem for some behavior which works, but without knowing why? It's easy to verify that the sequence \(x_n=\alpha\cdot\phi^n\) satisfies the equation \(x_{n+2}=x_{n+1}+x_n\): that's exactly how we calculate \(\phi\). However just from looking at the recurrence it's not obvious to use a geometric sequence. In this case the ansatz is a hint that allows you to solve this problem without too much linear algebra. In other cases, it might be a trick that has been proved successfully elsewhere. My discomfort was related to the fact that I probably couldn't motivate the ansatz enough. However you seem to estimate the post all the same. Cheers Thomas 

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