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Fibonacci Sequences

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The Fibonacci Sequence is a recursively defined sequence determined by the function:

Fn = 0 if n = 0
Fn = 1 if n = 1
Fn-2 + Fn-1 if n ≥ 2

where n is a natural number.

The first few terms of the sequence are:
F0 = 0, F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, F6 = 8, F7 = 13, F8 =21 .....

Consider the formula ( Fn)^2 + ( Fn+1 )^2 = F2n+1.

a) Show that the formula is true for n = 0 and n = 1.
b) Prove the formula is true for all natural numbers n by induction.

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Fibonacci Sequences are investigated.

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