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Thread: Am I on the right path to this induction problem?

  1. #1
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    Am I on the right path to this induction problem?

    Professor gave us this problem to work at home after lecture ended. I believe I am right. Then i just have to manipulate algebra at the end.
    http://imgur.com/gallery/wC11kX1
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  2. #2
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    Re: Am I on the right path to this induction problem?

    Quote Originally Posted by math951 View Post
    Professor gave us this problem to work at home after lecture ended. I believe I am right. Then i just have to manipulate algebra at the end.
    http://imgur.com/gallery/wC11kX1
    So suppose that $\sum\limits_{k = 1}^N {{k^3}} = {\left( {\frac{{N(N + 1)}}{2}} \right)^2}$ is true

    Then we look at $\sum\limits_{k = 1}^{N+1} {{k^3}}$
    $\begin{align*}\sum\limits_{k = 1}^{N + 1} {{k^3}} &= \sum\limits_{k = 1}^N {{k^3}} + {(N + 1)^3}\\
    &= {\left( {\frac{{N(N + 1)}}{2}} \right)^2} + {(N + 1)^3}\\
    &= \left( {\frac{{{N^2}{{(N + 1)}^2}}}{4}} \right) + {(N + 1)^3}\\
    &={(N + 1)^2}\left\{ {\left( {\frac{{{N^2}}}{4}} \right) + (N + 1)} \right\}\\
    &= {(N + 1)^2}\left( {\frac{{{N^2} + 4N + 4}}{4}} \right)\\
    &= {(N + 1)^2}\left( {\frac{{{{(N + 2)}^2}}}{4}} \right)\\
    &= {\left( {\frac{{(N + 1)(N + 2)}}{2}} \right)^2}\end{align*}$
    Last edited by Plato; Jul 10th 2019 at 02:04 PM.
    Thanks from topsquark and math951
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