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Math Help - induction

  1. #1
    Newbie
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    induction

    hi there,

    I'm struggling with the last bit of this induction question...

    Q: Let x>-1. Prove by induction that:

    (1+x)^n > 1+nx

    for every integer n>1
    --------
    So far I've got:

    Let P(n) be the statement (1+x)^n > 1+nx

    (1+x)^1 > 1+nx = 1+x

    Therefore P(1) is true.

    Assume true for some n+1

    I cannot figure out where to go from here..... could you possibly help me?

    Cheers
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  2. #2
    MHF Contributor arbolis's Avatar
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    tip

    Assume true for some n+1
    If I'm not wrong, it should be "Assume that p(n) is true, let's see if p(n)⇒p(n+1). "
    So you must start assuming that p(n) is right and you will be in need to use the p(n) formula. If it implies p(n+1), then p(n+1) is right, then p(n) is right for all n >1.
    In other words, using the formula (1+x)^n > 1+nx, you must reach (1+x)^{n+1}\ge1+(n+1)x.
    If you have some problem, ask us.
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  3. #3
    Eater of Worlds
    galactus's Avatar
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    I will skip the case for n=1.

    Induction hypothesis and show true for P(k+1).

    We have to show that (x+1)^{k+1}\geq{1+(k+1)x}

    (1+x)^{k}(1+x)\geq{(1+kx)(1+x)}

    (1+x)^{k+1}\geq{1+x+kx+kx^{2}}

    (1+x)^{k+1}\geq{1+(k+1)x+kx^{2}}

    The right side is clearly greater than 1+(k+1)x as we need.

    P(k+1) is true and the induction is complete.
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