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Math Help - mathematical induction

  1. #1
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    mathematical induction

    Can you please help me with
    Prove that n^3-n is divisible by 3
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  2. #2
    Super Member craig's Avatar
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    Quote Originally Posted by wantanswers View Post
    Can you please help me with
    Prove that n^3-n is divisible by 3
    n^3 - n = 3a for some a.

    First we see if it holds for n=0

    0^3 - 0 = 0, therefore true for n=0

    Now we use what's called the induction step, we assume that it's true for n=k where k is some natural nubmer.

    This gives us k^3 - k = 3a

    We then prove this for n=k+1, giving us

    (k+1)^3 - k - 1.

    Expanding the brackets gives k^3 + 3k^2 + 3k + 1 - k - 1

    k^3 + 3k^2 + 2k which can be written as k^3 + 3k^2 + 3k - k

    (k^3-k) + 3(k^2 + k) = 3a + 3(k^2 + k).

    Therefore it is true for k+1 if true for k.

    You know it is true for 0, therefore true for 1. True for one, therefore true for 2....

    Hope this helps
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