Why does ∑ k = 1 n 1 4 ( 1 4 k − 3 − 1 4 k + 1 ) turn into 1 4 ( 1 − 1 4 n + 1 )

Question

Why does       ∑          k      =      1        n        1    4    (      1          4      k      −      3        −      1          4      k      +      1        ) turn into       1    4    (  1  −      1          4      n      +      1        )

hofyonlines5 · Accepted Answer

∑                      k            =            1                    n                (                              1                          4              k              −              3                                      −                              1                          4              k              +              1                                      )            is a &quot;telescoping series&quot;. Write it out explicitly, and note most of the terms cancel.                                          ∑                          k              =              1                        n                    (                                    1                              4                k                −                3                                                        −                                              1                              4                k                +                1                                              )                            =                  (                                    1              1                                            −                                          1                5                                              )                    +                    (                                    1              5                                                        −                                                      1                9                                              )                    +                    (                                    1              9                                                        −                                                      1                13                                              )                    +                    ⋯                    +                    (                                    1                              4                n                −                7                                                                      −                                                      1                                  4                  n                  −                  3                                                              )          +                    (                                    1                              4                n                −                3                                                        −                                              1                              4                n                +                1                                              )                                          =                              1            −                                          1                                  4                  n                  +                  1                                                              .

Cory Patrick · Accepted Answer

∑          k      =      1        n    (      1          4      k      −      3        −      1          4      k      +      1        )  =  (      1    1    −      1    5    )  +  (      1    5    −      1    9    )  +  ⋯  +  (      1          4      n      −      7        −      1          4      n      −      3        )  +  (      1          4      n      −      3        −      1          4      n      +      1        )  =      1    1    −  (      1    5    −      1    5    )  −  (      1    9    −      1    9    )  −  ⋯  −  (      1          4      n      −      3        −      1          4      n      −      3        )  −      1          4      n      +      1        =  1  −      1          4      n      +      1      The (*) line should be       1    4    (  1  −      1          4      n      +      1        ), not       1    4    (  1  −      1          4      k      +      1        )

Why does <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> k =

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