In the proof of corollary 3.1.4.1, how did they write fn+1=α^(n+1)-β^(n+1) ( I know I didn't write the denominators, but I just don't understand this step). Is there a lot of algebra that they left out here? Or were they using something besides the Binet Formula? I think that the Lemmas 3.1.4.2-4 are interesting because they tell us properties of the Fibonacci numbers that will help us to actually use them to prove things, and just some basic properties so that we can work with them a little bit better (and it's just cool that these things are actually true).
Tuesday, October 11, 2011
2nd half of 3.1.4, due October 11
First, doing the polynomial division and looking at the patterns last class was really helpful and I'm surprised that the book didn't explain that part.
In the proof of corollary 3.1.4.1, how did they write fn+1=α^(n+1)-β^(n+1) ( I know I didn't write the denominators, but I just don't understand this step). Is there a lot of algebra that they left out here? Or were they using something besides the Binet Formula? I think that the Lemmas 3.1.4.2-4 are interesting because they tell us properties of the Fibonacci numbers that will help us to actually use them to prove things, and just some basic properties so that we can work with them a little bit better (and it's just cool that these things are actually true).
In the proof of corollary 3.1.4.1, how did they write fn+1=α^(n+1)-β^(n+1) ( I know I didn't write the denominators, but I just don't understand this step). Is there a lot of algebra that they left out here? Or were they using something besides the Binet Formula? I think that the Lemmas 3.1.4.2-4 are interesting because they tell us properties of the Fibonacci numbers that will help us to actually use them to prove things, and just some basic properties so that we can work with them a little bit better (and it's just cool that these things are actually true).
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