Se ( n +1/n)² = 3 então n³ + 1/n³ vale ?.... Pergunta de ideia deTourugo

May 2020 1 83 Report

Se ( n +1/n)² = 3 então n³ + 1/n³ vale ?

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$\left(n+\frac{1}{n}\right)^2=3\\\\\left(n+\frac{1}{n}\right)=\sqrt{3}\\\\\left(n+\frac{1}{n}\right)^3=(\sqrt{3})^3\\\\n^3+\frac{3n^2}{n}+\frac{3n}{n^2}+\frac{1}{n^3}=\sqrt{27}\\\\n^3+3n+\frac{3}{n}+\frac{1}{n^3}=3\sqrt{3}\\\\n^3+3\left(n+\frac{1}{n}\right)+\frac{1}{n^3}=3\sqrt{3}\\\\n^3+3\sqrt{3}+\frac{1}{n^3}=3\sqrt{3}\\\\\boxed{n^3+\frac{1}{n^3}=0}$

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