Binet Form/First Form

Theorem
Let $m \in \R$.

Define:

The recursive sequence:
 * $U_n = m U_{n - 1} + U_{n - 2}$

where:

has the closed-form solution:
 * $U_n = \dfrac {\alpha^n - \beta^n} \Delta$

Proof
Proof by induction:

Let $\map P n$ be the proposition:
 * $U_n = \dfrac {\alpha^n - \beta^n} \Delta$

Basis for the Induction
We have:

Therefore $\map P 0$ and $\map P 1$ are true.

This is the basis for the induction.

Induction Hypothesis
This is our induction hypothesis:
 * For some $k \in \N$, both $\map P k$ and $\map P {k + 1}$ are true.

That is:
 * $U_k = \dfrac {\alpha^k - \beta^k} \Delta$
 * $U_{k + 1} = \dfrac {\alpha^{k + 1} - \beta^{k + 1}} \Delta$

Now we need to show true for $n = k + 2$:
 * $\map P {k + 2}$ is true.

That is:
 * $U_{k + 2} = \dfrac {\alpha^{k + 2} - \beta^{k + 2}} \Delta$

Induction Step
This is our induction step:

First we notice that:

Similarly:

Thus:

This show that $\map P {k + 2}$ is true.

By principle of mathematical induction, $\map P n$ is true for all $n \in \N$.