Relation Between First and Second Form of Binet Form
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Theorem
Let $m \in \R$.
Define:
\(\ds \Delta\) | \(=\) | \(\ds \sqrt {m^2 + 4}\) | ||||||||||||
\(\ds \alpha\) | \(=\) | \(\ds \frac {m + \Delta} 2\) | ||||||||||||
\(\ds \beta\) | \(=\) | \(\ds \frac {m - \Delta} 2\) |
First Form
The recursive sequence:
- $U_n = m U_{n - 1} + U_{n - 2}$
where:
\(\ds U_0\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds U_1\) | \(=\) | \(\ds 1\) |
has the closed-form solution:
- $U_n = \dfrac {\alpha^n - \beta^n} \Delta$
Second Form
The recursive sequence:
- $V_n = m V_{n - 1} + V_{n - 2}$
where:
\(\ds V_0\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds V_1\) | \(=\) | \(\ds m\) |
has the closed-form solution:
- $V_n = \alpha^n + \beta^n$
where $\Delta, \alpha, \beta$ are as for the first 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:
\(\ds U_0 + U_2\) | \(=\) | \(\ds 0 + m U_1 + U_0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m \times 1 + 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds V_1\) | Definition of $V_1$ | |||||||||||
\(\ds U_1 + U_3\) | \(=\) | \(\ds 1 + m U_2 + U_1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m \times m + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m \times V_1 + V_0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds V_2\) | Definition of $V_2$ |
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:
\(\ds V_{k + 2}\) | \(=\) | \(\ds m V_{k + 1} + V_k\) | Definition of Recursive Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds m \paren {U_k + U_{k + 2} } + \paren {U_{k - 1} + U_{k + 1} }\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {m U_k + U_{k - 1} } + \paren {m U_{k + 2} + U_{k + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds U_{k + 1} + U_{k + 3}\) |
This show that $\map P {k + 2}$ is true.
By Principle of Mathematical Induction, $\map P n$ is true for all $n \in \N _{>0}$.
$\blacksquare$
Source of Name
This entry was named for Jacques Philippe Marie Binet.
Sources
- Weisstein, Eric W. "Binet Forms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinetForms.html