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Math Help - Induction Help!!

  1. #1
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    Induction Help!!

    Question
    Prove by induction that n in set positive

    1**3 + 2**3 + 4**3 ... n**3 = n**2(n+1)**2/4

    base case n = 1

    n**(n+1)**2/4

    1**2(1+1)**2/4

    1(2)**2/4

    4/4 = 1 so true

    inductive step n=k

    n=k+1

    1**3 + 2**3 + 4**3 ... k**3 = k**2(k+1)**2/4

    k**2(k+1)**2/4 + (k+1)**3 = (k+1)**2 (k+1+1)**2/4

    k**2(4+1)**2/4+ 4(k+1)**3/4 = (k**2+1)(k+2)**2/4

    Am i on the right track ?
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  2. #2
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    Tejas
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    Re: Induction Help!!

    Your post is extremely hard to read. What I think you are trying to prove is:

    $\displaystyle \sum_{i = 1}^n i^3 = \dfrac{n^2(n+1)^2}{4}$

    It is clear this holds for $n = 1$.

    Assuming that:

    $\displaystyle \sum_{i = 1}^k i^3 = \dfrac{k^2(k+1)^2}{4}$ for a given $k$, we have:

    $\displaystyle \sum_{i = 1}^{k+1} i^3 = \sum_{i = 1}^k i^3 + (k+1)^3 = \dfrac{k^2(k+1)^2}{4} + (k+1)^3$

    $= \dfrac{(k+1)^2}{4}[k^2 + 4(k+1)]$

    and you may continue from here. You are not supposed to just "replace" $k$ with $k+1$ and see if it works.
    Thanks from Odail
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  3. #3
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    Re: Induction Help!!

    Thanks much appreciated
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