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Some Algebra Identities
04-29-2010, 11:01 AM
Post: #1
Some Algebra Identities
(1) $(\sum_{k=1}^n y_k)\sum_{k=1}^n x_k^2/y_k - (\sum_{k=1}^n x_k)^2 = \sum_{i=1}^n \sum_{j=1}^n y_i x_j^2/ y_j-(\sum_{k=1}^n x_k)^2$
$= \displaystyle{\sum_{k=1}^n x_k^2 + \sum_{i \neq j} y_i x_j^2/ y_j -\sum_{k=1}^n x_k^2 - \sum_{1<=i<j<=n} 2x_i x_j = \sum_{1<=i<j<=n} (y_i x_j^2/ y_j + y_j x_i^2/ y_i - 2x_i x_j)}$
$= \displaystyle{\sum_{1<=i<j<=n} (x_i y_j - x_j y_i)^2 / (y_i y_j)}$
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