# Cassini's Identity/Proof 1

## Theorem

$F_{n + 1} F_{n - 1} - F_n^2 = \paren {-1}^n$

## 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:

 $\ds F_{k + 2} F_k - F_{k + 1}^2$ $=$ $\ds \left({F_k + F_{k + 1} }\right) F_k - F_{k + 1}^2$ $\ds$ $=$ $\ds F_k^2 + F_k F_{k + 1} - F_{k + 1}^2$ $\ds$ $=$ $\ds F_k^2 + F_k F_{k + 1} - F_{k + 1} \left({F_k + F_{k - 1} }\right)$ $\ds$ $=$ $\ds F_k^2 + F_k F_{k + 1} - F_k F_{k + 1} - F_{k + 1} F_{k - 1}$ $\ds$ $=$ $\ds F_k^2 - F_{k + 1} F_{k - 1}$ $\ds$ $=$ $\ds \left({-1}\right) \left({F_{k + 1} F_{k - 1} - F_k^2}\right)$ $\ds$ $=$ $\ds \left({-1}\right) \left({-1}\right)^k$ $\ds$ $=$ $\ds \left({-1}\right)^{k + 1}$

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

$\blacksquare$

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.

## Source of Name

This entry was named for Giovanni Domenico Cassini.