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Pythagorean Fibs

Age 16 to 18
Challenge Level

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.