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'Pythagorean Fibs' printed from https://nrich.maths.org/
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.