Cassini's Identity/Proof 1

Proof
We see that:
 * $F_2 F_0 - F_1^2 = 1 \times 0 - 1 = -1 = \left({-1}\right)^1$

so the proposition holds for $n = 1$.

We also see that:
 * $F_3 F_1 - F_2^2 = 2 \times 1 - 1 = \left({-1}\right)^2$

so the proposition holds for $n = 2$.

Suppose the proposition is true for $n = k$, that is:
 * $F_{k + 1} F_{k - 1} - F_k^2 = \left({-1}\right)^k$

It remains to be shown that it follows from this that the proposition is true for $n = k + 1$, that is:
 * $F_{k + 2} F_k - F_{k + 1}^2 = \left({-1}\right)^{k + 1}$

So:

By the Principle of Mathematical Induction, the proof is complete.

Note that from the above we have that:
 * $F_{k + 2} F_k - F_{k + 1}^2 = \left({-1}\right)^{k + 1}$

from which:
 * $F_{n + 1}^2 - F_n F_{n + 2} = \left({-1}\right)^n$

follows immediately.