Does this equation hold ? ∑ i N ∑ j N ∑ k N ( A i − A j ) ( A i − A k ) = C ∑ i N ∑ j N ( A i − A j ) 2

Question

Does this equation hold ?      ∑          i              N            ∑          j              N            ∑          k              N        (      A    i    −      A    j    )  (      A    i    −      A    k    )  =  C      ∑          i              N            ∑          j              N        (      A    i    −      A    j        )    2

blogrom1j7hf8 · Accepted Answer

First, introduce some notation:  S  =  ∑      A    l    T  =  ∑      A    l    2    .Now, the left hand side: do the innermost sum first, to get      ∑    i        ∑    j    (      A    i    −      A    j    )  (  N      A    i    −  S  )  .Do the next inner sum, to get      ∑    i    (  N      A    i    −  S      )    2    .Finally, open the square to get      N    2    T  −  2  N      S    2    +  N      S    2    =      N    2    T  −  N      S    2  On the right hand side.Do the inner sum to get      ∑    i    (  N      A    i    2    −  2      A    i    S  +  T  )  .Do the remaining sum to get  N  T  −  2      S    2    +  N  T  =  2  N  T  −  2      S    2    ,

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