Proving a properties of fibonacci numbers

[John112]

Let p[n] be the following property of Fibonacci numbers:
p[n]: f_{n+1} * f_{n-1} - (f_{n})^{2} = (-1)^{n}. Prove p[n] by induction.

This is the proof I wrote. i used regular induction. Is weak induction sufficient to prove it or do I need to prove this by strong induction?

proof:

BASE STEP: p[1]: f_{2} * f_{0} - (f_{1})^{2} = (1 * 0) - 1 = -1

and (-1)^{1} = -1

INDUCTIVE STEP:

assume P[k]: f_{k+1} * f_{k-1} - (f_{k})^{2} = (-1)^{k}

show p[k+1]: f_{k+2} * f_{k} - (f_{k+1})^{2} = (-1)^{k+1}
We know f_{k+2} = f_{k+1} + f_{k}

p[K+1]: f_{k+2} * f_{k} - (f_{k+1})^{2} = (-1)^{k+1}

substitute f_{k+2} = f_{k+1} + f_{k} into p[k+1]

p[k+1]: (f_{k+1} + f_{k}) * f_{k} - (f_{k+1})^{2} = (-1)^{k+1}

= f_{k} * f_{k+1} + (f_{k})^{2} - (f_{k+1})^{2}

We know p[k]: f_{k+1} * f_{k-1} - (f_{k})^{2} = (-1)^{k}

We can rewrite p[k] as: (f_{k})^{2} = f_{k+1} * f_{k-1} - (-1)^{k}

We substitue (f_{k})^{2} from p[k] into the (f_{k})^{2} in p[k+1] = f_{k+1} + f_{k} + f_{k+1} * f_{k-1} - (-1)^{k} - f_{k+1}^{2})

= f_{k+1} (f_{k+1} + f_{k}) (-1)^{k} - f_{k+1}^{2})

= (f_{k+1})^{2}) - (-1)^{k} - (f_{k+1})^{2})

= - ((-1)^{k})

= (-1)^{k+1}Does this proof strong enough or would I have to do this by strong induction? If by strong induction, how do I got about proving it?

 

[mfb]

Every valid proof with weak induction is a valid proof with strong induction, too (and this is the first time I saw specific words for them). Your proof uses valid steps only, therefore it is valid. There is no need to do more.