Cassini's Identity/Proof 1

Theorem
Let $F_k$ be the $k$th Fibonacci number.

Then $F_{n+1}F_{n-1} - F_n^2 = \left({-1}\right)^n$.

This is also sometimes reported (slightly less elegantly) as $F_{n+1}^2 - F_n F_{n+2} = \left({-1}\right)^n$

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.