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.