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I need to prove the rule below using Mathematical proof by induction but actually I'm stuck in the middle of the proving.

$$F_n^2 - F_{n-1}F_{n+1} = (-1)^{n+1} {~\rm for~} n>1$$

If someone can help that would be great.

Thank you

Christopher Carl Heckman
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Dinuki Seneviratne
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  • I'm assuming you're stuck on the inductive part. If you want to show that $$F_{k+1}^2-F_k F_{k+2}=(-1)^{k+2},$$ notice that you have a $F_{k+2}$ on the left hand side which you should get rid of. Use the identity $F_{k+2}=F_{k+1}+F_k$, multiply out, look for something to replace, and then simplify your answer. (You'll have to use the Fibonacci definition again, if I remember correctly.) – Christopher Carl Heckman Apr 25 '16 at 04:05
  • @Christopher Carl Heckman. Thank you for the help. :) – Dinuki Seneviratne Apr 25 '16 at 04:09

1 Answers1

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Note that $$F_{n+1} = F_n+F_{n-1}\quad\mbox{for all }n\ge2,$$ which follows that \begin{align} F_{n+1}^2 - F_{n}F_{n+2} &=F_{n+1}(F_{n}+F_{n-1}) - F_{n}(F_{n+1}+F_{n}) \\ &=F_{n+1}F_{n-1}-F_{n}^2\\ &= -(-1)^{n+1}\\ &=(-1)^{n+2}. \end{align}

Solumilkyu
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