Parts (2) and (3) use the earlier parts of the question.

For part (3), as usual try small values of n first, look for a pattern and make a conjecture about the result you expect might always be true.

To prove your conjecture take

F_n=b-a, F_n+1=a, F_n+2 = b, F_n+3 = b+a

because this symmetry in the algebra will make the working simpler.