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Math Help - Mathematical Induction

  1. #1
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    Mathematical Induction

    What is wrong with this "proof"?

    "Theorem" For every positive integer n, \sum_{i=1}^n  i = (n+1/2)^2/2

    Basic step : The formula is true for n = 1

    Inductive Step: Suppose that \sum_{i=1}^n  i = (n+\frac{1}{2})^2/2. Then \sum_{i=1}^{n+1}  i = (n+\frac{1}{2})^2/2 = (\sum_{i=1}^n  i) + (n+1) . By the inductive hypothesis, \sum_{i=1}^{n+1}  i = (n+\frac{1}{2})^2/2 + n + 1 = (n^2 + n + \frac{1}{4})/2 + n + 1 = (n^2 + 3n + \frac{9}{4})/2 = (n + \frac{3}{2})^2/2 = [(n+1) + \frac{1}{2}]^2/2, completing the inductive step.

    What I have found wrong is the basic step actually doesn't hold true for n = 1. But I cant find any other mistakes
    Last edited by Discrete; July 16th 2007 at 05:53 PM.
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  2. #2
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    The sum of the first n integers is \frac{n(n+1)}{2}, not \frac{(n+\frac{1}{2})^{2}}{2}

    The induction should show that your theroem is false.

    Since it doesn't hold for 1, that's a clue.
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  3. #3
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    Quote Originally Posted by galactus View Post
    The sum of the first n integers is \frac{n(n+1)}{2}, not \frac{(n+\frac{1}{2})^{2}}{2}

    The induction should show that your theroem is false.

    Since it doesn't hold for 1, that's a clue.
    is that all the mistakes then besides the basic step and the one you mentioned?
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