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Prove: $$F^2_{n+1} - F_nF_{n+2} = (-1)^n$$

I do understand where does $$F^2_{k+2} - F_{k+1}F_{k+3} = $$ come from. But then it distributes as follows: $$=F_{k+2}(F_{k+1} + F_k)-F_{k+1}(F_{k+2} + F_{k+1})=$$ $$=F_{k+2}F_k - F^2_{k+1}=-(F^2_{k+1}-F_kF_{k+2})=-(-1)^k=(-1)^{k+1}$$ I see no pattern. Can someone explain this transition

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    $\begingroup$ It is the Fibonacci sequence! Hence, $F_{n+2} = F_{n+1}+F_{n}$ for all $n \ge 0$. $\endgroup$ – cos_dm_math21 Mar 20 at 13:01
  • $\begingroup$ @cos_dm_math21 Okay. It seems obvious only now, but i still don't understand where $F_{k+2}F_k-F^2_{k+1}$ comes from $\endgroup$ – kertal Mar 20 at 13:14
  • $\begingroup$ $F_{k+2}(F_{k+1}+F_k) - F_{k+1}(F_{k+2}+F_{k+1}) = F_{k+2}F_{k+1} + F_{k+2}F_k - F_{k+1}F_{k+2} - F_{k+1}F_{k+1} = F_{k}F_{k+2} - F_{k+1}^2$, and you know that $F_{k+1}^2 - F_kF_{k+2} = (-1)^k$ from the induction hypothesis. $\endgroup$ – cos_dm_math21 Mar 20 at 13:21
  • $\begingroup$ @cos_dm_math21 Appreciate a lot. You're brilliant (: $\endgroup$ – kertal Mar 20 at 13:30