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FP1 Proof By Induction, stuck on induction step watch

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    Hello
    I'm currently doing one question of chapter 6 of FP1, but I am stuck of how to simply this down.
    The question is:
    Prove by Mathematical Induction that:
    \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2

    Now I have been able to do the basic step and the assumption step. I started the induction step when n=k+1 and I got to the stage where I had to simplify the equation:

    \frac{1}{4}k^2(k+1)^2 +(k+1)^3

    Now is there a way of simplifying this expression, for I tried to use the internet and the textbook, but it doesn't seem to explain it. Sorry if this seems to be a silly question. Thanks
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    (Original post by cpdavis)
    Hello
    I'm currently doing one question of chapter 6 of FP1, but I am stuck of how to simply this down.
    The question is:
    Prove by Mathematical Induction that:
    \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2

    Now I have been able to do the basic step and the assumption step. I started the induction step when n=k+1 and I got to the stage where I had to simplify the equation:

    \frac{1}{4}k^2(k+1)^2 +(k+1)^3

    Now is there a way of simplifying this expression, for I tried to use the internet and the textbook, but it doesn't seem to explain it. Sorry if this seems to be a silly question. Thanks
    Take out a factor of \frac{1}{4}(k+1)^2
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    (Original post by cpdavis)
    Hello
    I'm currently doing one question of chapter 6 of FP1, but I am stuck of how to simply this down.
    The question is:
    Prove by Mathematical Induction that:
    \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2

    Now I have been able to do the basic step and the assumption step. I started the induction step when n=k+1 and I got to the stage where I had to simplify the equation:

    \frac{1}{4}k^2(k+1)^2 +(k+1)^3

    Now is there a way of simplifying this expression, for I tried to use the internet and the textbook, but it doesn't seem to explain it. Sorry if this seems to be a silly question. Thanks
    take a factor of 0.25(k+1)^2 out,
    youll get 0.25(k+1)^2 (k^2 +4(k+1))
    and it turns out to factorise to 0,25(k+1)^2 (k+2)^2 which is the form you want,

    hope it helps (:
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    As everyone else has said....

    However, look at the form of the result you are trying to achieve. You can see that n^2 is a factor, hence you would be looking for (k+1)^2 as a factor in the induction step.
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    Thanks Everyone! Repping everything before this post!
 
 
 
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