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Thread: Prove by induction? I think...

  1. #1
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    Prove by induction? I think...

    I am very lost on this problem and am not sure where to begin.. Any help would be much appreciated!

    Prove the Following:

    $\displaystyle sum_{i=0}^n i^3$ = $\displaystyle sum_{i=0}^n (i)^2$
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  2. #2
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    $\displaystyle \sum_{i=0}^2 i^3=0^3+1^3+2^3=1+8=9$

    $\displaystyle \sum_{i=0}^2 i^2=0^2+1^2+2^2=1+4=5$

    am I missing something? seems like disproof by counter example
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  3. #3
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    Quote Originally Posted by vexiked View Post
    Prove the Following:
    $\displaystyle sum_{i=0}^n i^3$ = $\displaystyle sum_{i=0}^n (i)^2$
    Surely you mean $\displaystyle \sum\limits_{k = 1}^n {k^3 } = \left( {\sum\limits_{k = 1}^n k } \right)^2 ?$

    Yes do it by induction. But simplify the right hand sum first.
    $\displaystyle \sum\limits_{k = 1}^n k = \frac{{n(n + 1)}}{2}$ so that $\displaystyle \left( {\sum\limits_{k = 1}^n k } \right)^2 = \left( {\frac{{n(n + 1)}}{2}} \right)^2 = \frac{{n^2 (n + 1)^2 }}{4}$.

    Now do your induction.
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  4. #4
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    Yes this is what I meant.. Thanks alot! I have the worst time trying to compute closed form solutions.... maybe I just do it incorrectly
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