Proving Algebraic expression involving a SummationI'm following a derivation and am stumped by one of the steps: ∑ k = 1 13 ( 1 − k − 1 13 ) 3 = 1 13 3 ∑ k = 1 13 k 3 I am stumped as to how ( 14 − ( k − 1 ) ) 3 = k 3 Any help advice appreciated

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Proving Algebraic expression involving a SummationI&#039;m following a derivation and am stumped by one of the steps:      ∑          k      =      1              13        (  1  −            k      −      1        13        )    3    =      1          13      3            ∑          k      =      1              13            k    3  I am stumped as to how  (  14  −  (  k  −  1  )      )    3    =      k    3  Any help advice appreciated

Emma Cooper · Accepted Answer

Because it&#039;s      1          13      3            ∑          k      =      1              13        (  14  −  k      )    3    =      1          13      3            ∑          k      =      1              13            k    3  Just      13    3    +      12    3    +  .  .  .  +      2    3    +      1    3    =      1    3    +      2    3    +  .  .  .  +      12    3    +      13    3    .

Daniel Paul · Accepted Answer

your equation            (      1      −                        k          −          1                13            )        3    =      k    3  is not true, form here would we get  1  −            k      −      1        13    =  kand so  13  −  k  +  1  =  13  k

Proving Algebraic expression involving a Summation I'm following a derivation and am stumped by one of the steps: sum_(k=1)^(13)(1 - (k-1)/(13))^3 = 1/(13^3)sum_(k=1)^(13)k^3 I am stumped as to how (14 - (k-1))^3= k^3

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