Prove of the Cassini-Samson Identity
The Cassini-Samson Identity is an identity that involves the Fibonnaci numbers, it states that Fn−1Fn+1−F2n=(−1)n for n∈N.
We thus prove the theorem by induction:
The base case n=1 is easily verified.
Next, assume n=k and prove for n=k+1:
FkFk+2−F2k+1
=Fk(Fk+Fk+1)−F2k+1
=F2k+Fk+1(Fk−Fk+1)
=F2k−Fk+1Fk1=−(−1)k=(−1)k+1. Which is thus proved.
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