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Math Help - Induction Inequality

  1. #1
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    Induction Inequality

    Prove by induction
    (1+h)^n≥ 1 + nh + ((n(n-1))/2)h^2
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  2. #2
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    is your h positive??

    tried something, hope it helps

    The inequality is true for n=0 (easy to show)

    Assume the inequality holds for n

    (1+n)^{n+1}
    =(1+h)(1+h)^{n}
    \ge (1+h)(1+nh+\frac{n(n-1)h^{2}}{2})
    =1+nh+\frac{n(n-1)h^{2}}{2}+h+nh^{2}+\frac{n(n-1)h^{3}}{2}
    =1+(n+1)h+\frac{n(n+1)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}
    \ge 1+(n+1)h+\frac{n(n+1)h^{2}}{2} if h is positive
    Last edited by acc100jt; September 10th 2008 at 08:46 PM.
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  3. #3
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    I'm a little confused as to how you went from the third line from the bottom to the next line.
    Last edited by Snooks02; September 10th 2008 at 09:12 PM.
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  4. #4
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    =1+nh+\frac{n(n-1)h^{2}}{2}+h+nh^{2}+\frac{n(n-1)h^{3}}{2}
    =1+nh+h+\frac{(n^{2}-n)h^{2}}{2}+\frac{2nh^{2}}{2}+\frac{n(n-1)h^{3}}{2}
    =1+(n+1)h+\frac{(n^{2}-n+2n)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}
    =1+(n+1)h+\frac{(n^{2}+1)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}
    =1+(n+1)h+\frac{n(n+1)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}
    \ge 1+(n+1)h+\frac{n(n+1)h^{2}}{2} if h is positive

    is this ok now?
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