You remember the Fibonacci numbers... Using the principle of mathematical induct

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let us prove this by induction base case : n=2 F3 =F1+F2 =2 now F1F3-F2^2 = 1*2-1^2 =2-1 = 1 = (-1)^2 So base case is true Let us assume that this is true for n => F(n-1) F(n+1) - Fn^2 =(-1)^n ----------- (1) => we know that F(n+1) = F(n) + F(n-1) (from given formula) => F(n-1) = F(n+1) - F(n) Substituting this in equation(1) we get (F(n+1) - F(n)) F(n+1) - Fn^2 =(-1)^n => F(n+1)^2 - F(n)F(n+1) - Fn^2 =(-1)^n =>F(n+1)^2 - F(n)(F(n+1) +Fn) =(-1)^n =>F(n+1)^2 - F(n)(F(n+2)) =(-1)^n multipling both sides with -1 we get F(n)F(n+2) - F(n+1)^2 = (-1)^n+1 Hence the equation is true for n+1 by induction this is true for all n Hence proved

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