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Thread: reciprocals

  1. December 7th 2007, 06:51 PM #1
    mr.toronto
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    reciprocals

    Are there 21 different positive whole numbers such that the sum of their reciprocals is 1?

    I'm not sure about this question
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  2. December 7th 2007, 10:07 PM #2
    badgerigar
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    Yes

    Yes.

    1/2+1/3 = 5/6
    1/2+1/3+\frac{1/2+1/3}{6} = 35/36
    1/2+1/3+\frac{1/2+1/3}{6}+\frac{\frac{1/2+1/3}{6}}{6} = \frac{6^3-1}{6^3}

    We can keep doing this forever so the sum of any even number of reciprocals of natural numbers can represent \frac{6^n-1}{6^n}
    Adding a last number of 1/6^n produces 1, so the sum of any odd number of reciprocals of 1atural numbers can be 1
    Last edited by badgerigar; December 7th 2007 at 10:08 PM. Reason: forgot math tags
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