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Math Help - principle of mathematical induction

  1. #1
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    principle of mathematical induction

    How do I prove by induction that n^2 is greater than or equal to 3n + 4 for n > 3?
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    Re: principle of mathematical induction

    First prove it for n=4.

    Next show that if it's true n=k then it's true for n=k+1.

    But I suppose you know that already...so why not say how far you got and where you're stuck.
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    Re: principle of mathematical induction

    Quote Originally Posted by a tutor View Post
    First prove it for n=4.

    Next show that if it's true n=k then it's true for n=k+1.

    But I suppose you know that already...so why not say how far you got and where you're stuck.
    I've gotten to 'assume true for n = k' and I dont know how to go about proving that its true for k + 1
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    Re: principle of mathematical induction

    Well, if it's true for n=k then k^2\ge 3k+4

    What does that tell you about (k+1)^2?
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    Re: principle of mathematical induction

    Quote Originally Posted by a tutor View Post
    Well, if it's true for n=k then k^2\ge 3k+4

    What does that tell you about (k+1)^2?
    (k+1)^2 is greater than or equal to 3(k+1) + 4?
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    Re: principle of mathematical induction

    OK, (k+1)^2=k^2+2k+1.

    Now, since k^2\ge3k+4

    we have

    (k+1)^2=

    k^2+2k+1\ge 5k+5=3(k+1)+2k+2= 3(k+1)+4+2k-2\ge 3(k+1)+4
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    Re: principle of mathematical induction

    The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
    1. The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
    2. The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.

    The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.
    Last edited by kalwin; May 24th 2012 at 04:42 AM.
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    Re: principle of mathematical induction

    Quote Originally Posted by kalwin View Post
    The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
    1. The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
    2. The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.

    The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.
    next time, just provide the link ...

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