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Math Help - Induction help

  1. #1
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    Cool Induction help

    Hi I have an induction question. I think I have the thing done properly, I'm just not 100% sure.

    The question is: prove that (3n)! <= 27^n (n!)^3
    for all positive integers n.

    This is what I have so far:

    show for 1:
    6<27

    assume true for k where k is some positive integer
    (3k)! <= 27^k (k!)^3

    prove true for k+1
    (3(k+1))! <= 27^(k+1) (k+1!)^3
    (3k + 3)! <= (27^k)(27)((k!)^3)((k+1)^3)
    ((3k)!)(3k+1)(3k+2)(3k+3) <= (27^k)(27)((k!)^3)((k+1)^3)
    THIS IS THE STEP I'M UNSURE ABOUT!!
    I take out the original equation.. is this ok?
    (3k+1)(3k+2)(3k+3) <= (27)(k+1)^3
    27k^3 + 54k^2 + 33k + 6 <= 27k^3 + 81m^2 + 81m + 27

    QED?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by pikminman View Post
    Hi I have an induction question. I think I have the thing done properly, I'm just not 100% sure.

    The question is: prove that (3n)! <= 27^n (n!)^3
    for all positive integers n.

    This is what I have so far:

    show for 1:
    6<27

    assume true for k where k is some positive integer
    (3k)! <= 27^k (k!)^3

    prove true for k+1
    (3(k+1))! <= 27^(k+1) (k+1!)^3
    (3k + 3)! <= (27^k)(27)((k!)^3)((k+1)^3)
    ((3k)!)(3k+1)(3k+2)(3k+3) <= (27^k)(27)((k!)^3)((k+1)^3)
    THIS IS THE STEP I'M UNSURE ABOUT!!
    I take out the original equation.. is this ok?
    (3k+1)(3k+2)(3k+3) <= (27)(k+1)^3
    27k^3 + 54k^2 + 33k + 6 <= 27k^3 + 81m^2 + 81m + 27

    QED?
    Note that \lim_{n\to\infty}\frac{27(n+1)!^3}{(3n+3)!}\cdot\f  rac{(3n)!}{27n!^3}=\frac{1}{27}. This means that \frac{27(n!)^3}{(3n)!}\to0 so clearly at some point (3n)!>27(n!)^3. Maybe you want to relook at the question.
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    Note that \lim_{n\to\infty}\frac{27(n+1)!^3}{(3n+3)!}\cdot\f  rac{(3n)!}{27n!^3}=\frac{1}{27}. This means that \frac{27(n!)^3}{(3n)!}\to0 so clearly at some point (3n)!>27(n!)^3. Maybe you want to relook at the question.
    I said 27^n not just 27. Maybe I don't understand what you mean.
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