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Math Help - mathematical induction

  1. #1
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    mathematical induction

    Here is the problem:

    Prove, by induction, that the sum (-1)^k (2k+1) from k=1 to 2n is proportional to n, and find the constant of proportionality.

    Well I don't know how to start the induction part but I have a hunch that the constant of proportionality is just going to be 2.

    Thank you
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  2. #2
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    Quote Originally Posted by Percent09 View Post
    Here is the problem:

    Prove, by induction, that the sum (-1)^k (2k+1) from k=1 to 2n is proportional to n, and find the constant of proportionality.

    Well I don't know how to start the induction part but I have a hunch that the constant of proportionality is just going to be 2.

    Thank you
    The base case is trivial (everything is proportional to 1), so let n = 2. Then
    \sum_{k = 1}^{4} (-1)^k (2k + 1) = -(2(1) + 1) + (2(2) + 1) - (2(3) + 1) + (2(4) + 1)  = -3 + 5 - 7 + 9 = 4 \propto 2

    So assume that the theorem is true for some n = N. Then we need to show that it is true for n = N + 1. Let the constant of proportionality be 2, since this is true of the n = 2 case.
    \sum_{k = 1}^{2N + 2} (-1)^k (2k + 1) = \sum_{k = 1}^{2N} (-1)^k (2k + 1) - (2(N + 1) + 1) + (2(N + 2) + 1)

    = 2N + 2 = 2(N + 1)

    This is proportional to N + 1 with a constant of proportionality 2. So etc, etc.

    -Dan
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