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Math Help - induction

  1. #1
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    induction

    how would I go about proving:

    the sum of the first n odd numbers is equal to n^2 for every n which is greater than or equal to 1.

    Thanks
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  2. #2
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    Quote Originally Posted by daaavo View Post
    how would I go about proving:

    the sum of the first n odd numbers is equal to n^2 for every n which is greater than or equal to 1.

    Thanks
    First comes the n = 1 case.
    \sum_{i = 1}^{1}(2i - 1) = 1 = 1^2 (check!)

    Now let the theorem be true for some n = k. We need to show that it is also true for n = k + 1.
    \sum_{i = 1}^{k + 1}(2i - 1) = \sum_{i = 1}^{k}(2i - 1) + (2(k + 1) - 1)

    By our assumption:
    \sum_{i = 1}^{k}(2i - 1) = k^2

    So
    \sum_{i = 1}^{k + 1}(2i - 1) = k^2 + 2(k + 1) - 1

    For your theorem to be true this must be equal to (k + 1)^2. Can you finish this?

    -Dan
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