(50g) Nth Fibonacci Number

[Thomas Klemm]

(02-27-2015 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