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    How do I prove that,

    \displaystyle\sum_{k=1}^n (4k^3-12k) = n(n+1)(n+3)(n-2)

    I have done this:

    4 \displaystyle\sum_{k=1}^n k^3 -12 \displaystyle\sum_{k=1}^n k

    4 \frac{1}{4}n^2(n+1)^2 -12\frac{1}{2}n(n+1)

    n^2(n+1)^2 -6n(n+1)

    Thanks.
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    Take n(n+1) out as a factor and then simplify the quadratic that you're left with.
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    Following on from yours
    n(n+1)(n(n+1)-6)
    n(n+1)(n^2+n-6)
    n(n+1)(n+3)(n-2)
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    (Original post by doivid)
    Following on from yours
    n(n+1)(n(n+1)-6)
    n(n+1)(n^2+n-6)
    n(n+1)(n+3)(n-2)
    How did you got rid of the minus?
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    (Original post by lnx)
    How did you got rid from the minus?
    Are you reading n-2 as n+2? As far as I can tell, no-one got rid of a minus (or needed to get rid of a minus).
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    (Original post by nuodai)
    Are you reading n-2 as n+2? As far as I can tell, no-one got rid of a minus (or needed to get rid of a minus).
    lol nah, but

    n^2(n+1)^2 -6n(n+1)

    if you divide this by n(n+1), wouldnt yuo get n(n+1)-6? :confused:
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    (Original post by lnx)
    lol nah, but

    n^2(n+1)^2 -6n(n+1)

    if you divide this by n(n+1), wouldnt yuo get n(n+1)-6? :confused:
    Yes... which is what doivid said
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    (Original post by nuodai)
    Yes... which is what doivid said
    How did he get: n(n+1)(n(n+1)-6)
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    (Original post by lnx)
    How did he get: n(n+1)(n(n+1)-6)
    I really can't see why you're confused here You have n^2(n+1)^2 - 6n(n+1) right? I'll go through it in extremely small steps:

    So:
    \newline n^2(n+1)^2 - 6n(n+1)\newline

= nn(n+1)(n+1) - 6n(n+1)\newline

= n(n+1)n(n+1) - 6n(n+1)\newline

= n(n+1) \Big[ n(n+1) - 6 \Big]\newline

= n(n+1)(n^2 + n - 6)\newline

= n(n+1)(n+3)(n-2)
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    (Original post by lnx)
    How did he get: n(n+1)(n(n+1)-6)

    expand the brackets

    n(n+1) * n(n+1) gives you your n^2(n+1)^2 and n(n+1) * -6 gives you your -6n(n+1).
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    (Original post by lnx)
    if you divide this by n(n+1), wouldnt yuo get n(n+1)-6? :confused:
    (Original post by lnx)
    How did he get: n(n+1)(n(n+1)-6)
    :confused:
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    haha ah okay, sorry guys for the confusion and thanks noudai for the steps.
 
 
 
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