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Math Help - help use induction to prove

  1. #1
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    Unhappy help use induction to prove

    use induction to prove

    13 + 23 + 33 + . . . + n3 = (1 + 2 + 3 + . . . + n)2
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    Re: help use induction to prove

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  3. #3
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    Re: help use induction to prove

    Quote Originally Posted by harryisland View Post
    use induction to prove

    13 + 23 + 33 + . . . + n3 = (1 + 2 + 3 + . . . + n)2
    I am going to assume that we are aware of the identity
    \sum_{i = 1}^k = \frac{k(k + 1)}{2}

    It is not necessary, but useful.

    First, does this hold for n = 1? Obviously.

    So assume that there exists a n = k such that
    1^3 + 2^3 + ~...~ + k^3 = (1 + 2 + ~...~ + k)^2

    So what can we do with n = k + 1?
    (1^3 + 2^3 + ~...~ + k^3) + (k + 1)^3  = ((1 + 2 + ~...~ + k) + (k + 1))^2

    By our assumption the first k terms on the LHS can be replaced by ( )^2. We are also going to expand the RHS as a square:
    (1 + 2 + ~...~ + k)^2 + (k + 1)^3  = (1 + 2 + ~...~ + k)^2 + 2(1 + 2 + ~...~ + k)(k + 1) + (k + 1)^2

    The first terms on both sides cancel, leaving
    (k + 1)^3  = 2(1 + 2 + ~...~ + k)(k + 1) + (k + 1)^2

    I'm going to leave the rest to you, but with the reminder of the sum above:
    \sum_{i = 1}^k = 1 + 2 + ~...~ + k = \frac{k(k + 1)}{2}

    If you need clarification, please let us know.

    -Dan
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