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Mathematical induction proof

  1. Nov 25, 2011 #1
    Hello everybody,
    I am doing my reading lately to prepare for some exams to join a mathematics department.

    And I would very much like, if anyone could help, the solution (or a hint) to the following induction proof.

    " Show that n^3 + (n+1)^3 + (n+2)^3 is a multiple of 9 "

    :smile:

    I can deal with proofs that say " *this* is divisible with *that* " but in that case I have no clue how to start.

    I have noone to ask for help ... but you in the forum.
     
  2. jcsd
  3. Nov 25, 2011 #2

    Mentallic

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    Show it's true for the base case n=1, then assume it's true for n=k, that is

    [tex]k^3+(k+1)^3+(k+2)^3 = 9M[/tex]

    where M is an integer, now prove it's true for n=k+1,

    [tex](k+1)^3+(k+2)^3+(k+3)^3=9N[/tex]

    So what you need to do is show that N is an integer as well. You'll need to make a substitution from the equation you assumed to get anywhere.
     
  4. Nov 26, 2011 #3
    Thanks for the answer,

    Okay, here is my solution. I thought that at the problem defintion "is divisible by" and "is a multiple of" makes a huge difference.

    The base case is obvious.

    Let that the statement holds for n=k, that is,
    [tex]k^3+(k+1)^3+(k+2)^3[/tex] is a multiple of 9.
    We will then show that it holds for n=k+1, that is [tex](k+1)^3+(k+2)^3+(k+3)^3[/tex] is a multiple of 9.

    We have,
    [tex]k^3+(k+1)^3+(k+2)^3+(k+3)^3-k^3=9M+(k+3)^3-k^3[/tex],
    So what we want to show is that [tex](k+3)^3-k^3[/tex] is also a multiple of 9.
    By using the formula [tex]a^3-b^3=(a-b)(a^2+b^2+ab)[/tex] we can reach our goal.
     
  5. Nov 26, 2011 #4

    Mentallic

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    Nicely done. The only thing I would add is to specifically state that the equation you assumed is equal to 9M, rather than telling us in words that it's a multiple of 9. This way there is no disputing anything when you make the substitution.
     
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