Proving the Evenness of Fibonacci Numbers through Division by 3

[mattmns]

Here is the question:
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Prove that $f_{n}$ is even if and only if n is divisible by 3. ($f_{n}$ is of course the nth Fibonacci number)
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Proving that n is divisible by 3 => $f_{n}$ is even is easily done by induction, but the other implication is eluding me. It is easy to show that $f_{n}$ is even iff $f_{n-3}$ is even, but I can't see if this helps. Any ideas about how to prove this implication? Thanks.

 

[slearch]

It is easy to show that $f_{n}$ is even iff $f_{n-3}$ is even, but I can't see if this helps.

If this is easy to show, then it would be enough to look at the evenness of $$f_0, f_1$$, and $$f_2$$.

 

[mattmns]

I was thinking about that, and I think you are absolutely right. Thanks.