Induction Proof with Fibonacci How do I prove this?    
For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.

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*Base Case:


For $n = 1$:
$$f_2^2 - f_2f_1 - f_1^2 = (-1)^1$$
$$(1)^2 - (1)(1) - (1)^2 = -1$$
$$-1 = -1$$



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*Induction Step:


Assume that the given statement is true. We now try to prove that it holds true for $n+1$:
$$f_{n+2}^2 - f_{n+2}f_{n+1} - f_{n+1}^2 = (-1)^{n+1}$$
Typically you choose one side and try to get to the other side. I will choose the left side:
$$f_{n+2}^2 - f_{n+2}f_{n+1} - f_{n+1}^2 = (f_{n+1} + f_{n})^2 - (f_{n+1} + f_{n})(f_{n+1}) - f_{n+1}^2$$
$$= f_{n+1}^2 + 2f_{n+1}f_{n} + f_{n}^2 - f_{n+1}^2 - f_{n+1}f_{n} - f_{n+1}^2$$
$$= f_{n}^2 + f_{n+1}f_{n} - f_{n+1}^2$$
$$= (-1)(f_{n+1}^2 - f_{n+1}f_{n} - f_{n}^2)$$
$$= (-1)(-1)^n$$
$$= (-1)^{n+1}$$
