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Math Help - sum of non-arithmetic,non-geometric series

  1. #1
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    sum of non-arithmetic,non-geometric series

    Find the sum of the series:
    (a) (1^2)+(3^2)+(5^2)+...+(2n-1)^2
    (b) (1^2)-(2^2)+(3^2)-(4^2)+...+((-1)^(n+1))(n^2)
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  2. #2
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    Quote Originally Posted by chris86 View Post
    Find the sum of the series:
    (a) (1^2)+(3^2)+(5^2)+...+(2n-1)^2
    (b) (1^2)-(2^2)+(3^2)-(4^2)+...+((-1)^(n+1))(n^2)
    Sum of the Squares of the first n Natural Numbers
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by chris86 View Post
    Find the sum of the series:
    (a) (1^2)+(3^2)+(5^2)+...+(2n-1)^2
    (b) (1^2)-(2^2)+(3^2)-(4^2)+...+((-1)^(n+1))(n^2)
    a)

    \displaystyle 1^{2} + 3^{2} + 5^{2} + ...+ (2n-1)^{2} =

    \displaystyle = 1^{2} + 2^{2} + 3^{2} + ...+ (2n-1)^{2} - 2^{2}\ \{1^{2} + 2^{2} + 3^{2} + ...+ (n-1)^{2}\} =

    \displaystyle = \frac{2n\ (2n-1)\ (4n-1) - 4n\ (n-1)\ (2n-1)}{6} =

    \displaystyle = \frac{2n\ (2n-1)\ (2n+1)}{6}

    Kind regards

    \chi \sigma
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