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Thread: Can you help me?

  1. #1
    Junior Member
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    Can you help me?

    Prove that 3^n > n^3 for all positive integers n.

    I first tried n= 1, which showed 3>1
    n= 2, 9>8
    but n = 3 became 27 = 27, which is not true, but everything else works afterwards.

    I was given the hint to use induction. So I tried this:

    3^(n+1) > (n+1)^3
    3*(3^n) > n^3 + 3n^2 + 3n + 1

    What do I do next?
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  2. #2
    Member
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    Suppose $\displaystyle 3^n > n^3$ for some $\displaystyle n = k, k \ge 4$ which was shown.

    For the inductive step, you want to show that $\displaystyle 3^{k+1} > (k+1)^3$.

    Well $\displaystyle (k+1)^3 = k^3 + 3k^2 + 3k + 1$.

    Since it is known that $\displaystyle k \ge 4$ then $\displaystyle k^3 \ge 4k^2 > 3k^2$.

    Also, $\displaystyle k \ge 4$ leads to $\displaystyle k^2 \ge 16$, so $\displaystyle k^2 - 3 \ge 13$ and it follows that $\displaystyle k(k^2-3) \ge 52 > 1$. Rearranging, it is evident that $\displaystyle 3k + 1 < k^3$.

    So applying the above revelations, $\displaystyle k^3 + 3k^2 + 3k + 1 = (k^3) + (3k^2) + (3k+1)$
    $\displaystyle < k^3 + k^3 + k^3 = 3k^3.$

    It is hypothesized that $\displaystyle k^3 < 3^k$, so $\displaystyle 3k^3 < 3 \cdot 3^k = 3^{k+1}.$

    Thus, $\displaystyle (k+1)^3 < 3^{k+1}$, and by induction, $\displaystyle 3^n > n^3$ is true for all $\displaystyle n \ge 4$.
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  3. #3
    Junior Member
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    Thank you for your help. But I need a little more explantion.


    I don't quite know where or why you did this particular line: k^3 >= 4k^2>3k^2

    Then why did you do the two lines after the above line?
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  4. #4
    Member
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    San Diego
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    I'll try to be clearer:

    I wanted to make it clear that $\displaystyle k^3 > 3k^2$ and $\displaystyle k^3 > 3k+1$.

    Using both of these facts, then $\displaystyle k^3+k^3 > 3k^2 + 3k +1$, so $\displaystyle k^3+k^3+k^3 > k^3 + 3k^2 + 3k +1$ which is the expansion of $\displaystyle (k+1)^3$.

    So all of the above sets up:

    $\displaystyle (k+1)^3 < 3k^3$

    and since $\displaystyle k^3 < 3^k$, then $\displaystyle 3k^3 < 3 \cdot 3^k$. in other words, $\displaystyle (k+1)^3 < 3^{k+1}$, which is what we wre trying to show.
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