I am having a lot of trouble figuring this problem out. Any help would be appreciated.
We can define a generalized Fibonacci sequence F1, F2, F3, F4, … by first selecting four integers a, b, c, and e, and then letting F1=a, F2=b, and Fn=cFn-1+eFn-2 if n>2.
a) Prove that, if d=gcd(a,b), then d|Fn for all n>=1.
b) Prove that, if f=gcd(Fm, Fm-1) and gcd(f, e)=1 then f|d.