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Math Help - smallest integer inequality

  1. #1
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    smallest integer inequality

    find the smallest positive integer n satisfying the inequality: 1+2+3+....+n>2007

    whats an inequality?

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  2. #2
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    Quote Originally Posted by mich13 View Post
    find the smallest positive integer n satisfying the inequality: 1+2+3+....+n>2007

    whats an inequality?

    xxxxxxxx
    Add them up,

    \frac{n(n+1)}{2} > 2007

    Now solve.
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  3. #3
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    Hello, mich13!

    Find the smallest positive integer n satisfying the inequality: . 1 + 2 + 3 + \cdots + n\;>\;2007

    whats an inequality? . . . . You don't know?
    The symbol {\bf{\color{blue}>}} means "greater than".


    You may be expected to know this formula . . .

    The sum of the positive integers from 1 to n is: . \frac{n(n+1)}{2}

    So we have: . \frac{n(n+1)}{2}\;> \;2007\quad\Rightarrow\quad n^2 + n \;>\;4014

    We have a quadratic inequality: . n^2 + n - 4014\;>\:0


    The equation: . n^2 + n - 4014 \:=\:0

    . . has roots: . n \;=\;\frac{-1 \pm\sqrt{1^2 - 4(1)(4014)}}{2(1)}\;=\;\frac{-1\pm\sqrt{16057}}{2} \;=\;\begin{Bmatrix}\;62.658... \\ \text{-}64.739...\end{Bmatrix}

    Therefore: . \boxed{n\:\geq\:63}

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  4. #4
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    Exclamation thanks you but....

    Quote Originally Posted by Soroban View Post
    Hello, mich13!

    The symbol {\bf{\color{blue}>}} means "greater than".


    You may be expected to know this formula . . .

    The sum of the positive integers from 1 to n is: . \frac{n(n+1)}{2}

    So we have: . \frac{n(n+1)}{2}\;> \;2007\quad\Rightarrow\quad n^2 + n \;>\;4014

    We have a quadratic inequality: . n^2 + n - 4014\;>\:0


    The equation: . n^2 + n - 4014 \:=\:0

    . . has roots: . n \;=\;\frac{-1 \pm\sqrt{1^2 - 4(1)(4014)}}{2(1)}\;=\;\frac{-1\pm\sqrt{16057}}{2} \;=\;\begin{Bmatrix}\;62.658... \\ \text{-}64.739...\end{Bmatrix}

    Therefore: . \boxed{n\:\geq\:63}

    no i dont know! I am 13 years old and i have no idea how to solve this!! i found it difficult to understand your meathod, but i managed! how did you get 4014 though? please explain it! thanks a lot xxx
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  5. #5
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    Quote Originally Posted by mich13 View Post
    no i dont know! I am 13 years old and i have no idea how to solve this!! i found it difficult to understand your meathod, but i managed! how did you get 4014 though? please explain it! thanks a lot xxx
    2007 x 2 = 4014
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