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Math Help - Inequality Inductions

  1. #1
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    Inequality Inductions

    Hi,

    Could someone please explain how to do this:

    Show by Induction that:
    n^2 - 2n + 5 > 0 for all n>=1

    I have so far:

    Show true for n = 1
    Assume true for n = k
    Show true for n = k+1 (using assumption)
    So:
    k^2 + 2k + 5 -2k - 1 >0
    |---- >0 ---|

    ??

    Thanks.
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  2. #2
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    Quote Originally Posted by justmaths View Post
    Hi,

    Could someone please explain how to do this:

    Show by Induction that:
    n^2 - 2n + 5 > 0 for all n>=1

    I have so far:

    Show true for n = 1
    Assume true for n = k
    Show true for n = k+1 (using assumption)
    So:
    k^2 + 2k + 5 -2k - 1 >0
    |---- >0 ---|

    ??

    Thanks.
    True for n = 1.

    Assume true for n = k, that is, assume k^2 - 2k + 5 > 0.

    Show that it follows from the assumption that it's true for n = k + 1:

    (k + 1)^2 - 2(k + 1) + 5 = k^2 + 2k + 1 - 2k - 2 + 5 = (k^2 - 2k + 5) + 2k - 1

    and it should be clear how to finish things off.
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  3. #3
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    But thats the thing, I don't know how to finish things off. I only know how to start these off. Could you please expand on your response, please?
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  4. #4
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    Quote Originally Posted by justmaths View Post
    But thats the thing, I don't know how to finish things off. I only know how to start these off. Could you please expand on your response, please?
    Well, from the inductive assumption you know that k^2 - 2k + 5 > 0. And you also know that 2k - 1 > 0. Therefore ....

    And you should have examples to follow (in class notes and/or textbook) for setting out a proof by induction in a formal way ....
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  5. #5
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    Yes I understand now. Thanks. I would have notes but I happened to be not in class for that lesson. I understood everything until the part where there are two parts which are bigger than zero. Thanks.
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