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Math Help - Combinatorics - Sum of series

  1. #1
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    Combinatorics - Sum of series

    Sum the series 1^2+2^2+ \cdots+n^2 by observing that m^2 =2 \dbinom{m}{2} + \dbinom{m}{1} and using the identity  \dbinom{0}{k}+ \dbinom{1}{k} + \cdots + \dbinom{n}{k}= \dbinom{n+1}{k+1}.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tarheelborn View Post
    Sum the series 1^2+2^2+ \cdots+n^2 by observing that m^2 =2 \dbinom{m}{2} + \dbinom{m}{1} and using the identity  \dbinom{0}{k}+ \dbinom{1}{k} + \cdots + \dbinom{n}{k}= \dbinom{n+1}{k+1}.
    lets see what you've tried.
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    I am really not sure where to start with this. I did verify that m^2 = 2* \dbinom{m}{2}+ \dbinom{m}{1}. So 1^2 = 2* \dbinom{1}{2} + \dbinom{1}{1} = (2*0)+1 = 1, 2^2=2* \dbinom{2}{2}+ \dbinom{2}{1}=2(1)+2=4 and n^2=2* \dbinom{n}{2} + \dbinom{n}{1} = 2*\frac{n!}{2!(n-2)!} + \frac{n!}{1*(n-1)!}.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tarheelborn View Post
    I am really not sure where to start with this. I did verify that m^2 = 2* \dbinom{m}{2}+ \dbinom{m}{1}. So 1^2 = 2* \dbinom{1}{2} + \dbinom{1}{1} = (2*0)+1 = 1, 2^2=2* \dbinom{2}{2}+ \dbinom{2}{1}=2(1)+2=4 and n^2=2* \dbinom{n}{2} + \dbinom{n}{1} = 2*\frac{n!}{2!(n-2)!} + \frac{n!}{1*(n-1)!}.
    you're simplifying prematurely. slow down. just plug in the expressions first:

    \displaystyle 1^2 + 2^2 + \cdots + n^2 = 2 {1 \choose 2} + {1 \choose 1} + 2 {2 \choose 2} + {2 \choose 1} + \cdots + 2 {n \choose 2} + {n \choose 1}

    Now what? (you can write out the expression for a few more terms if you don't see a pattern or what to do)
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  5. #5
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    Combinatorics - sum of series

    I really just don't see it. It looks like it would be  \dbinom{n}{n+1} + \dbinom{n}{n} but that doesn't make sense.
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