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m Reverted 1 edit by 122.166.1.94 (talk) to last revision by D.Lazard |
→Decomposition of powers: Break up long equation |
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<math display=block>\varphi^n = F_n\varphi + F_{n-1}.</math>
This equation can be [[Mathematical proof|proved]] by [[Mathematical induction|induction]] on {{math|''n'' ≥ 1}}:
<math display=block>\begin{align}
\varphi^{n+1} &= (F_n\varphi + F_{n-1})\varphi = F_n\varphi^2 + F_{n-1}\varphi \\ &= F_n(\varphi+1) + F_{n-1}\varphi = (F_n + F_{n-1})\varphi + F_n = F_{n+1}\varphi + F_n. \end{align}</math>
For <math>\psi = -1/\varphi</math>, it is also the case that <math>\psi^2 = \psi + 1</math> and it is also the case that
<math display=block>\psi^n = F_n\psi + F_{n-1}.</math>
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