@angeliccanary Hi there! Thanks for these ideas. It's really cool when you can find more than one, or even several, ways of doing the same problem. You can definitely use the recursion definition of the Fibonacci sequence (which is very similar to what Prof. Loh did). I do think Prof. Loh was trying to find a different way to prove this pattern, rather than assuming the pattern was true, so I'm not sure about this:
or use the fact that we are finding GCD's of consecutive fibonacci numbers.