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Math Help - Induction

  1. #1
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    Induction

    By using mathematical induction:
    Show (n^4) − 12(n^2) + 25n − 17 > 0 for all integers n ≥ 2.
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  2. #2
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
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    Test it for n=2 to show its positive.

    Assume its >0 for n

    Set n=n+1 and expand the equation but dont add the parts together. This should give n^4 - 12n^2 + 25n - 17 + (4n^3 + 6n^2 - 20n + 14).

    Now do the same process on the part of this equation thats in brackets to prove its also positive, if it is, that means the entire equation above is positive. Hence positive for the inductive case n=2, n and n+1 hence +ve for all n \geq 2

    It is for n=2;
    Assume it is for n;
    Let n=n+1 and expand the equation to give 4n^3 + 6n^2 -20n + 14 + (12n^2 + 4n + 4) > 0 since we assumed 4n^3 + 6n^2 -20n + 14 > 0 and 12n^2 + 4n + 4 is clearly greater than 0.

    So its positive for the base case n=2, n and n+1, hence +ve for all n \geq 2

    Sorry i wrote this in a hurry if its unclear let me know.
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