Mock AIME 2 Pre 2005 Problems/Problem 2

Problem

$x$ is a real number with the property that $x+\tfrac1x = 3$ . Let $S_m = x^m + \tfrac{1}{x^m}$ . Determine the value of $S_7$ .

Solution

We can calculate $x^2 + \dfrac{1}{x^2} = \left(x + \dfrac{1}{x}\right)^2 - 2 = 3^2 -2 = 7.$ Similarly, $x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right) \left(x^2 + \dfrac{1}{x^2}\right) - \left(x + \dfrac{1}{x}\right) = 3 \cdot 7 - 3 = 18$ and $x^4 + \dfrac{1}{x^4} = \left(x^2 + \dfrac{1}{x^2}\right)^2 - 2 = 7^2 - 2 = 47.$ Finally, $x^7 + \dfrac{1}{x^7} = \left(x^3 + \dfrac{1}{x^3}\right) \left(x^4 + \dfrac{1}{x^4}\right) - \left(x + \dfrac{1}{x}\right) = 18 \cdot 47 - 3 = \boxed{843}.$

Mock AIME 2 Pre 2005 (Problems, Source)
Preceded by Problem 1	Followed by Problem 3
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Mock AIME 2 Pre 2005 Problems/Problem 2

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Solution

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