Understanding Fibonacci Proof

Question

I'm trying to show that $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2} ∀ ≥1$$ where $$F_k = F_{k-1} + F_{k-2}$$ with $$F_0 = F_1 = 1$$

Let P(n) = $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2}$$

Basic Step: is P(1) true?

$$(F^2_2 - F^2_1) = 3 = 3 = (F_0 * F_3)$$ Is true because $$LHS = RHS$$

Assume that $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2}$$ is true, then we want to show that P(k+1) is also true:

$$F^2_{k+2} - F^2_{k+1} = (F_{k+2} – F_{k+1})(F_{k+2} + F_{k+1})$$

$$= F_k * F_{k+3}$$

Thus, P(k) <----> P(k+1) ?

Answer 1

$$F_{n+2}=F_{n+1}+F_n$$

Set $n=k,k+1$ one by one.

Can you see the way, now?

Answer 2

No induction needed: $F_{k+1}^2-F_k^2=(F_{k+1}-F_k)(F_{k+1}+F_k)=F_{k-1}\cdot F_{k+2}$