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Thread: Simple Proof by Induction

  1. #1
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    Simple Proof by Induction

    Use induction to prove that for every integer $\displaystyle n \geq 4$, $\displaystyle 3^n > n^3 $.

    Here's the solution:



    I don't understand where the term $\displaystyle k^3+3k^2+4k$ came from in the middle line. Could anyone explain?
    Last edited by Roam; Oct 24th 2009 at 09:37 PM.
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  2. #2
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    Quote Originally Posted by Roam View Post
    Use induction to prove that for every integer $\displaystyle n \geq 4$, $\displaystyle 3^n > n^3 $.

    Here's the solution:



    I don't understand where the term $\displaystyle k^3+3k^2+4k$ came from in the middle line. Could anyone explain?
    Since $\displaystyle k \ge 4$, we know that $\displaystyle k^3 > 3k^2$ and also that $\displaystyle k^3 > 4k$. Therefore, $\displaystyle k^3+k^3+k^3 > k^3+k^3+4k > k^3+3k^2 + 4k \Rightarrow k^3 + k^3 + k^3 > k^3 + 3k^2 + 4k$
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