19 lines
856 B
HTML
19 lines
856 B
HTML
<!DOCTYPE html>
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<html>
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<head>
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<meta charset="utf-8">
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<title>JS Bin</title>
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</head>
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<body>
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base \( \sin x \) line
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The [[eigenvalue]]s of the matrix '''''A''''' are \(\scriptstyle \varphi=\frac12(1+\sqrt5)\,\!\) and \(\scriptstyle 1-\varphi=\frac12(1-\sqrt5)\), for the respective
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[[eigenvector]]s \(\scriptstyle {\varphi \choose 1}\) and \(\scriptstyle {1-\varphi \choose 1}\) .
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Since \(\scriptstyle {\varphi \choose 1} - \scriptstyle {1-\varphi \choose 1}= \sqrt{5} ~\scriptstyle {1\choose 0}\), and \(\vec F_{n} = \mathbf{A}^n \vec F_{0},\)
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the above closed-form expression for the {{mvar|n}}th element in the Fibonacci series as an [[analytic function]] of {{mvar|n}} is now read off directly,
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:\[ F_{n} = \cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^n~.\]
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</body> |