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Math Help - proof by induction with inequalities

  1. #1
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    proof by induction with inequalities

    question is prove by induction on m that m^3 <= 2^m for m=>10
    thats supposed to be m cube less than or equal to 2 to the power n for m greater than or equal to 10

    P(K)= k^3 less than or equal to 2^k
    P(K+1)= (k+1)^3 less than or equal to 2^k+1

    2k^3 less than or equal to 2^k+1

    (k+1)^3 less than or equal to 2k^3
    now im not sure to do next
    please help
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  2. #2
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    Re: proof by induction with inequalities

    \begin{align*} (k+1)^3&=k^3+3k^2+3k+1\\ &\le k^3+3k^2+3k^2+k^2&& \text{since }k\ge3 \\&=k^3+7k^2\\&\le k^3+k^3&& \mbox{since }k\ge7\\&\le2^k+2^k&& \text{by induction hypothesis}\\&=2^{k+1}\end{align*}

    A couple of remarks. For each claim it should be clear whether you are assuming it, trying to prove it or something else. Your proof is just a series of statements, and the role of those statements is not clear. Write comments when necessary, such as "We assume ...," "We need to prove ...," "It follows from ... that ..."

    Mathematics is case-sensitive, so K and k may denote different variables.

    Edit: Don't forget the base case.
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    Re: proof by induction with inequalities

    Quote Originally Posted by emakarov View Post
    \begin{align*} (k+1)^3&=k^3+3k^2+3k+1\\ &\le k^3+3k^2+3k^2+k^2&& \text{since }k\ge3 \\&=k^3+7k^2\\&\le k^3+k^3&& \mbox{since }k\ge7\\&\le2^k+2^k&& \text{by induction hypothesis}\\&=2^{k+1}\end{align*}
    thanks but i dont understand where the second line has come from, how did you get k^3+3k^2+3k^2+k^2
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    Re: proof by induction with inequalities

    Quote Originally Posted by narjsingh View Post
    thanks but i dont understand where the second line has come from, how did you get k^3+3k^2+3k^2+k^2
    1\le k^2
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    Re: proof by induction with inequalities

    Quote Originally Posted by Plato View Post
    1\le k^2
    im really really sorry i still dont get that im a bsc econ student
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    Re: proof by induction with inequalities

    Quote Originally Posted by narjsingh View Post
    im really really sorry i still dont get that im a bsc econ student
    Because k\le k^2~\&~1\le k^2 we get 3k+1\le 3k^2+k^2

    k^3+3k^2+3k+1\le k^3+3k^2+3k^2+k^2
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  7. #7
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    Re: proof by induction with inequalities

    Quote Originally Posted by Plato View Post
    Because k\le k^2~\&~1\le k^2 we get 3k+1\le 3k^2+k^2

    k^3+3k^2+3k+1\le k^3+3k^2+3k^2+k^2
    thank you so much! i am really getting it! one last hing how did you get k is greater than or equal to 7?
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    Re: proof by induction with inequalities

    Quote Originally Posted by narjsingh View Post
    thank you so much! i am really getting it! one last hing how did you get k is greater than or equal to 7?
    3+3+1=7 so 3k^2+3k^2+k^2=7k^2
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