Try to show:

a)

b) for every n.

Using these two properties you can get:

. This is either 1 or 2, since Fn and F_n-1 are coprime. So you already know that the only possible values are 1 and 2.

To show that they are equal to what you right, just nottice the pattern 0,0,1,0,0,1,.... when looking to the sequence Fn (or Ln) modulo 2.