text: # Help with Fibonacci Identity
  
  ## Main Question or Discussion Point
  
  Can someone guide me on how to prove that
  $$F_{4n+3} + F_{4n+6} = F_{2n+1}^2 + F_{2n+4}^2$$
  
  either side of the above is the difference
  
  $$(F_{2n+2}*F_{2n+3} + F_{2n+4}^2) - (F_{2n}*F_{2n+1} + F_{2n+2}^2)$$
  
  I intend to post this sequence $$F_{2n}*F_{2n+1} + F_{2n+2}^2$$, with a comment re a few properties thereof, on Sloane's online encyclopedia of integer sequences but would like to verify the above identity first.
  
  Related Linear and Abstract Algebra News on Phys.org
  First thing springs to mind is to try use general formula for nth
  Fibbonacci number:
  
  $$F_{n}=\frac{\phi^n-(1-\phi)^n}{\sqrt{5}}$$
  
  In order to proceed with math induction.But I'm unsure will it work or not.
  I'm sure there are better methods ,though.
  
  Last edited:
  First thing springs to mind is to try use general formula for nth
  Fibbonacci number:
  
  $$F_{n}=\frac{\phi^n-(1-\phi)^n}{\sqrt{5}}$$
  
  In order to proceed with math induction.But I'm unsure will it work or not.
  I'm sure there are better methods ,though.
  Thanks
  I think there is an identity for the following that works:
  
  $$F_{i}*F_{j} + F_{i+1}*F_{j+1} = F_{?}$$
  
  Let j = i = 2n+1 then
  
  $$F_{2n+1}^{2} + F_{2n+2}^{2} = F_{4n+3}$$
  [Tex]F_{2n+2}^{2} + F_{2n+3}^{2} = F_{4n+5}[/tex]
  [Tex]F_{2n+2}^{3} + F_{2n+4)^{2} = F_{4n+7}[/tex]
  \\
  [Tex]F_{4n+3} +F_{4n+6} = F_{4n+3} + F_{4n+7} - F_{4n+5}[/tex]
  [Tex] =F_{2n+1}^{2} + F_{2n+4}^{2}[/tex]
  
  Last edited:
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