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Thread: Examples of Identity of Finite Sum of Integers

  1. #1
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    Examples of Identity of Finite Sum of Integers

    Hello.

    If someone know interesting examples of identity of finite sum of integers such as $\displaystyle \textstyle 70^2 = \sum_{k=1}^{24} k^2 $, could you tell me some please.

    Thank you.

    Misako Kawasoe
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    How about $\displaystyle \sum_{i=1}^{n} i{n \choose i} = n2^{n-1} $.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chiph588@ View Post
    How about $\displaystyle \sum_{i=1}^{n} i{n \choose i} = n2^{n-1} $.
    How about $\displaystyle \sum_{j=1}^{n}j^2{j\choose n}=n(2n+1)2^{n-2}$?
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    $\displaystyle 3^2+4^2=5^2$

    $\displaystyle 3^3+4^3+5^3=6^3$

    $\displaystyle 1^3+2^3+\dots+n^3=(1+2+\dots+n)^2$
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Drexel28 View Post
    How about $\displaystyle \sum_{j=1}^{n}j^2{j\choose n}=n(2n+1)2^{n-2}$?
    Or... $\displaystyle \sum_{j=1}^{n}j^3{j\choose n}=n^2(n+3)2^{n-3}$
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chiph588@ View Post
    Or... $\displaystyle \sum_{j=1}^{n}j^3{j\choose n}=n^2(n+3)2^{n-3}$
    Haha. Or! Define $\displaystyle f_1(x)=n(1+x)^{n-1}$ and define $\displaystyle f_{m+1}(x)=\left(x\cdot f_m(x)\right)'$. Then, $\displaystyle \sum_{j=1}^{n}j^m{n\choose j}=f_m(1),\text{ }m\geqslant 1$
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  7. #7
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    Thank you so much, everyone!
    I really appreciate these replies.
    The Bruno J.-san's example fits my request best because I want a "particular" example rather than general one. I should have written in my first post $\displaystyle 70^2 = 1^2 + 2^2 + \dots + 24^2 $ without using sigma symbol...
    Anyway, thanks a lot!
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