Find the integer closest to a − bLet a = 1 2 1 + 2 2 3 + 3 2 5 + … + 1001 2 2001 .Let b = 1 2 3 + 2 2 5 + 3 2 7 + … + 1001 2 2003 .Then, what is the closest integer of ( a − b )?

Question

Find the integer closest to   a  −  bLet  a  =            1              2              1    +            2              2              3    +            3              2              5    +  …  +            1001              2              2001    .Let  b  =            1              2              3    +            2              2              5    +            3              2              7    +  …  +            1001              2              2003    .Then, what is the closest integer of   (  a  −  b  )?

Erzrivalef6 · Accepted Answer

We have                    a        −        b                            =                  ∑                      k            =            1                                1001                                                k            2                                2            k            −            1                          −                  ∑                      k            =            1                                1001                                                k            2                                2            k            +            1                                                            =                              1            2                    1                −                              1001            2                    2003                +                  ∑                      k            =            1                                1000                                                (            k            +            1                          )              2                        −                          k              2                                            2            k            +            1                                                            =                              1            2                    1                −                              1001            2                    2003                +                  ∑                      k            =            1                                1000                          1                                          =        1001        −                  1001                      2            +                          1              1001                                                                        ≈        1001        −                  1001          2                ⋅                  (          1          −                      1            2002                    )                                                  =        500                              3            4

Denisse Valdez · Accepted Answer

a        −        b        =                  ∑                      i            =            1                                1001                                                (                                                                                i                                                        2                                                                              2                i                −                1                                      −                                                                                i                                                        2                                                                              2                i                +                1                                      )                    =                          ∑                      i            =            1                                1001                                                              2                                                                    i                                                        2                                                                                      4                                                                    i                                                        2                                                              −              1                                =                          1          2                          ∑                      i            =            1                                1001                                                              4                                                                    i                                                        2                                                              −              1                                      4                                                                    i                                                        2                                                              −              1                                +                      1            2                                ∑                          i              =              1                                      1001                                                          1                              4                                                                            i                                                              2                                                                      −                1                                                                              a        −        b        =                  1001          2                +                  1          4                          ∑                      i            =            1                                1001                                                (                          1                              2                i                −                1                                      −                          1                              2                i                +                1                                      )                          =                  1001          2                +                  1          4                          (          1          −                      1            2003                    )

Find the integer closest to a − b Let a = 1 <mro

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