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Can you explain to me the following equalities: \begin{align} \sum_{i=1}^n \left[\left( \sum_{j=1}^n (x_i-x_j) \right)\left( \sum_{k=1}^n (x_i-x_k) \right) \right]= \sum_{i,j,k=1}^n (x_i - x_j)(x_i - x_k) = \sum_{k=1}^n \left[ \sum_{i,j=1}^n \tfrac 12 (x_i - x_j)^2 \right] \end{align}

More generally, what are other good “product of sum” formulas to know? For example, the diagonal/off-diagonal decomposition of a squared sum is useful.

dunno
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  • The first equality involves moving all the $\sum$ symbols to the left hand side and then rewriting them as a single symbol – Henry May 03 '18 at 10:56
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    And the first expression is $\displaystyle\sum_{i=1}^n \left[ \left(\sum_{j=1}^n (x_i-x_j) \right)^2 \right]$ while the third is $\displaystyle \frac{n}{2}\sum_{i=1}^n \sum_{j=1}^n \left[ (x_i-x_j) ^2 \right] $ – Henry May 03 '18 at 11:12
  • Thanks! Where does the $n/2$ come in though? I’m stuck on that – dunno May 04 '18 at 21:37
  • I get the $n$ so it’s just the $1/2$ that is giving me troubke – dunno May 04 '18 at 21:38

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