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Math Help - Arithmetic Series

  1. #1
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    Arithmetic Series

    1. Find the equation that gives the sum of the first n positive integer
    ^
    the answer given was n(n+1)/2 - but I don't know how to get it

    2. Show that the sum of the first n odd integers is equal to the perfect square n^2

    3. Show that the sum of the first n even integers is equal to n^2 + n
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  2. #2
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    Let n be a positive integer. We will write S_{n} the sum of the first n integers.

    1)
    1 + 2 + ... + n = S_{n}
    n + (n-1) + ... + 1 = S_{n}

    So if we sum these two equations, we get

    (n+1) + ((n-1)+2) + ... + (n+1) = 2S_{n} , that is to say

    (n+1) + (n+1) + ... + (n+1) = n(n+1) = 2S_{n}

    Therefore S_{n}=\frac{n(n+1)}{2}

    2)
    What are the first n odd integers? 1,3,...,2n-1
    So their sum is
    \sum\limits_{k=1}^{n}(2k-1)=\sum\limits_{k=1}^{n}2k-\sum\limits_{k=1}^{n}1=2\sum\limits_{k=1}^{n}k-n=n(n+1)-n=n(n+1-n)=n^{2}

    3)
    It's easier than 2)
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