I am trying to figure out how to show that f_(n+1)f_(n-1)-(f_n)^2=(-1)^n when n is a positive integer.

My Work

P(n) is f_(n+1)f_(n-1)-(f_n)^2=(-1)^n for n>=1

Basis step - P(1) is true because f_2f_0-(f_1)^2=1*0-1=-1=(-1)^1

Inductive step - Assume P(k)=f_(k+1)f_(k-1)-(f_k)^2=(-1)^k is true, then P(k+1)=f_(k+2)f_(k)-(f_(k+1))^2=(-1)^(k+1) is true

P(k+1)=[f_(k+1)f_(k-1)-(f_k)^2]+[f_(k+2)f_(k)-(f_(k+1))^2]=(-1)^k+[f_(k+2)f_(k)-(f_(k+1))^2]

I can't figure out how to get this equation to equal (-1)^(k+1)