Fibonacci sequence, division

  • Thread starter mattmns
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  • #1
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Here is the question:
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Prove that [itex]f_{n}[/itex] is even if and only if n is divisible by 3. ([itex]f_{n}[/itex] is of course the nth Fibonacci number)
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Proving that n is divisible by 3 => [itex]f_{n}[/itex] is even is easily done by induction, but the other implication is eluding me. It is easy to show that [itex]f_{n}[/itex] is even iff [itex]f_{n-3}[/itex] is even, but I can't see if this helps. Any ideas about how to prove this implication? Thanks.
 

Answers and Replies

  • #2
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mattmns said:
It is easy to show that [itex]f_{n}[/itex] is even iff [itex]f_{n-3}[/itex] 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 [tex]f_0, f_1[/tex], and [tex]f_2[/tex].
 
  • #3
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I was thinking about that, and I think you are absolutely right. Thanks.
 

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