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Equation Magic?!

Algebra Level 3

If \(x + \dfrac{1}{x} = 2\), what can be said about the value of \(x^n + \dfrac{1}{x^n}\) for any real number \(n \neq 1\)?

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The values are different for certain \(n\)'s. The resulting value must be 2 for the rest of \(n\)'s. The signs of the value 2 alternate.

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by Michael Huang

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