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Math Help - induction question

  1. #1
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    induction question

    I have to show the following by mathematical induction:

    for each n in the natural numbers and for a > -1, the following inequality holds
    (1 + a)^n >= 1 + na + (n/2)(n-1)a^2 + (n/6)(n-1)(n-2)a^3.

    I know how to do the base case and the induction step.

    For n + 1, the left hand side of the inequality is (1+a)^(n+1) = (1+a)(^n)*(1+a). Then I multiplied the right hand side of the inequality by (1+a) assuming that the "n" case is correct.

    After multiplying through, I get the following:
    1 + (n+1)a + (n/2)(n+1)a^2 + (n/6)(n+1)(n-1)a^3 + (n/6)(n-1)(n-2)a^4. The first four terms are precisely the original right hand side of the inequality in "n+1." (i.e. what I'm looking for). However, how do I know the left hand side is still greater than or equal to the right hand side with the additional term (n/6)(n-1)(n-2)a^3 under these conditions. Do you know what I'm saying? I think I did all the arithmetic and algebra correctly.

    Thanks for any help or suggestions.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    I have to show the following by mathematical induction:

    for each n in the natural numbers and for a > -1, the following inequality holds
    (1 + a)^n >= 1 + na + (n/2)(n-1)a^2 + (n/6)(n-1)(n-2)a^3.

    I know how to do the base case and the induction step.

    For n + 1, the left hand side of the inequality is (1+a)^(n+1) = (1+a)(^n)*(1+a). Then I multiplied the right hand side of the inequality by (1+a) assuming that the "n" case is correct.

    After multiplying through, I get the following:
    1 + (n+1)a + (n/2)(n+1)a^2 + (n/6)(n+1)(n-1)a^3 + (n/6)(n-1)(n-2)a^4. The first four terms are precisely the original right hand side of the inequality in "n+1." (i.e. what I'm looking for). However, how do I know the left hand side is still greater than or equal to the right hand side with the additional term (n/6)(n-1)(n-2)a^3 under these conditions. Do you know what I'm saying? I think I did all the arithmetic and algebra correctly.

    Thanks for any help or suggestions.
    It looks vaguely like a binomial expansion of some sort. Are you sure that it isn't -1 \leq a \leq 1? If that was the case then your result follows easily. Else if we have a large enough positive a then
    \frac{n(n - 1)(n - 2)}{6}a^4 > \frac{n(n + 1)(n - 1)}{6}a^3
    for any n \geq 2.

    -Dan
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  3. #3
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    It is related to the Bernoulli inequality. Maybe that page will help.
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