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Math Help - Summation Problem

  1. #1
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    Summation Problem

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    Ʃn i2 = n(n+1)(2n+1)/6, n>=1
    i=1

    Should I use the formula
    Ʃn i = n(n+1)/2
    i=1

    and if so, how would I go about doing this?
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  2. #2
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    Use induction.
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  3. #3
    Math Engineering Student
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    Quote Originally Posted by jaclyn91 View Post

    Should I use the formula
    Ʃn i = n(n+1)/2
    i=1
    Yes, this works.

    Now, use this simple fact: i(i+1)=\frac{1}{3}\left( i(i+1)(i+2)-i(i-1)(i+1) \right).

    So the sum becomes \sum\limits_{i=1}^{n}{i(i+1)}=\frac{1}{3}\sum\limi  ts_{i=1}^{n}{\left( i(i+1)(i+2)-i(i-1)(i+1) \right)}, now this is

    \underbrace{\sum\limits_{i=1}^{n}{i^{2}}}_{\text{Y  ou need this}}+\underbrace{\sum\limits_{i=1}^{n}{i}}_{\tex  t{You know this}}=\frac{1}{3}\underbrace{\sum\limits_{i=1}^{n  }{\left( i(i+1)(i+2)-i(i-1)(i+1) \right)}}_{n(n+1)(n+2),\text{ because the above telescopes}}.

    I think you can take it from there.
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