Assuming, (after confirming base case)

F_{k+2}F_{k}− (F_{(k+1)}^{2}= (−1)^{k}. (Assumption P)

With the induction step, you want to prove

F_{k+3}F_{k+1}- (F_{k+2})^{2}= (-1)^{k+1}

so,

(-1)^{k+1}= (-1)^{k}*(-1)^1 =-1 * ( F) (by Assumption P)_{k+2}F_{k}− (F_{(k+1)}^{2}

Can you take it from here? I currently don't have pencil and paper handy to give you the full answer.

You'll probably also have to use the fact that F_{k+2}= F_{k}+ F_{k+1}

For simplicity, I would leave the bolded and then work in the other direction as well

In other words, working with F_{k+3}F_{k+1}- (F_{k+2})^{2}so that it's equal to what is in the bold, since A = B = C implies A = C