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Assume that, for $n\ge1$,$$a_n=\sqrt{1+\left(1+\frac{1}{n}\right)^2 } +\sqrt{1+\left(1-\frac{1}{n}\right)^2 } $$

How to prove that

$$\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_{20}}$$ is an integer?

  • 4
    Rationalize the fractions, telescope.2012-08-25
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    on an unrelated note, the squareroot symbol looks terrible with the concave roof.2012-08-25
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    @progressiveforest Happy now?2012-08-25
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    I wonder if behind the problem there is a fast solution ...2012-08-25
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    @peter wow ok I thought it's an inherent problem with the math typesetting here but apparently it can be improved.2012-08-26

2 Answers 2