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Math Help - Induction Inequality

  1. #1
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    Induction Inequality

    Any help with the following proof would be appreciated,

    Determine for which natural numbers k^2 - 3k >= 4 and prove your answer.

    How would we use induction here? Thanks in advance for all the help!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Prove that:

    (i) The inequality is true for k=4 .

    (ii) If it is true for k\geq 4 integer, then it is true for k+1 .


    Fernando Revilla
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  3. #3
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    Quote Originally Posted by jstarks44444 View Post
    Any help with the following proof would be appreciated,

    Determine for which natural numbers k^2 - 3k >= 4 and prove your answer.

    How would we use induction here? Thanks in advance for all the help!
    kk-3k\ge\ 4\Rightarrow\ k(k-3)\ge\ 4

    k cannot be 3 or less.

    k=4\Rightarrow\ 4(1)=4

    Then we have that the inequality ought to be true for k\ge\ 4

    P(k)

    k(k-3)\ge\ 4

    P(k+1)

    (k+1)(k-2)\ge\ 4

    Try to show that "if" P(k) is true, "then" P(k+1) will also be true
    (establish the inductive "cause and effect")

    Proof

    (k+1)(k-2)=(k+1)(k-2-1)+(k+1)=(k+1)(k-3)+(k+1)

    =k(k-3)+(k-3)+(k+1)=k(k-3)+2k-2

    If P(k) is true, then the above is \ge\ 4+2k-2

    with k\ge\ 4

    and so P(k+1) is true also.
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