Relation Between First and Second Form of Binet Form

Theorem
Let $m \in \R$.

Define:

Second Form
For any given value of $m$:
 * $U_{n - 1} + U_{n + 1} = V_n$

Proof
Proof by induction:

Let $\map P n$ be the proposition:
 * $U_{n - 1} + U_{n + 1} = V_n$

Basis for the Induction
We have:

Therefore $\map P 1$ and $\map P 2$ are true.

This is the basis for the induction.

Induction Hypothesis
This is our induction hypothesis:
 * For some $k \in \N_{> 0}$, both $\map P k$ and $\map P {k + 1}$ are true.

That is:
 * $U_{k - 1} + U_{k + 1} = V_k$
 * $U_k + U_{k + 2} = V_{k + 1}$

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

That is:
 * $U_{k + 1} + U_{k + 3} = V_{k + 2}$

Induction Step
This is our induction step:

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

By Principle of Mathematical Induction, $\map P n$ is true for all $n \in \N _{>0}$.