Honsberger's Identity/Proof 2
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Theorem
- $\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
Proof
\(\ds \) | \(\) | \(\ds F_{m - 1} F_n + F_m F_{n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m - 1} - \hat \phi^{m - 1} } {\sqrt 5} \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} + \dfrac {\phi^m - \hat \phi^m} {\sqrt 5} \dfrac {\phi^{n + 1} - \hat \phi^{n + 1} } {\sqrt 5}\) | Euler-Binet Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} - \phi^{m - 1} \hat \phi^n - \phi^n \hat \phi^{m - 1} + \hat \phi^{m + n - 1} + \phi^{m + n + 1} - \phi^m \hat \phi^{n + 1} - \phi^{n + 1} \hat \phi^m + \hat \phi^{m + n + 1} } 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} \paren {1 + \phi^2} + \hat \phi^{m + n - 1} \paren {1 + \hat \phi^2} - \phi^{m - 1} \hat \phi^n \paren {1 + \phi \hat \phi} -\phi^n \hat \phi^{m - 1} \paren {1 + \phi \hat \phi} } 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} \paren {1 + \phi^2} + \hat \phi^{m + n - 1} \paren {1 + \hat \phi^2} } 5\) | as $\phi \hat \phi = -1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} \paren {2 + \phi} + \hat \phi^{m + n - 1} \paren {2 + \hat \phi} } 5\) | as both $\phi$ and $\hat \phi$ satisfy $x^2 = x + 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} \paren {2 + \dfrac {1 + \sqrt 5} 2} + \hat \phi^{m + n - 1} \paren {2 + \dfrac {1 - \sqrt 5} 2} } 5\) | Definition of $\phi$ and $\hat \phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} \paren {\dfrac {5 + \sqrt 5} 2} + \hat \phi^{m + n - 1} \paren {\dfrac{ 5 - \sqrt 5} 2} } 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n - 1} \paren {\dfrac {1 + \sqrt 5} 2} - \hat \phi^{m + n - 1} \paren {\dfrac {1 - \sqrt 5} 2} } {\sqrt 5}\) | dividing numerator and denominator by $\sqrt 5$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^{m + n} - \hat \phi^{m + n} } {\sqrt 5}\) | Definition of $\phi$ and $\hat \phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{m + n}\) | Euler-Binet Formula |
$\blacksquare$