Proof By Induction Fibonacci Numbers How do I prove that
$$ f_{ 2n+1 } = 3f_{ 2n } + 1 - f_{ 2n-3 } $$
I'm not sure how to prove it using the defining recurrence of Fibonacci numbers.

As pointed out in Golob's answer, your equation is not in fact true.  However we have
$$\eqalign{f_{2n+1}
  &=f_{2n}+f_{2n-1}\cr
  &=(f_{2n-1}+f_{2n-2})+f_{2n-1}\cr
  &=2f_{2n-1}+(f_{2n-1}-f_{2n-3})\cr}$$
and therefore
$$f_{2n+1}=3f_{2n-1}-f_{2n-3}\ .$$
Is there any possibility that this is what you meant?
