Solve the equation: S n = 12 ( 4 − 3 ) ( 4 2 − 3 2 ) + 12 2 ( 4 2 − 3 2 ) ( 4 3 − 3 3 ) + . . . + 12 n ( 4 n − 3 n ) ( 4 n + 1 − 3 n + 1 )

Question

Solve the equation:      S    n    =      12          (      4      −      3      )      (              4        2            −              3        2            )        +            12      2              (              4        2            −              3        2            )      (              4        3            −              3        3            )        +  .  .  .  +            12      n              (              4        n            −              3        n            )      (              4                  n          +          1                    −              3                  n          +          1                    )

candelo6a · Accepted Answer

12      n              (              4        n            −              3        n            )      (              4                  n          +          1                    −              3                  n          +          1                    )            4    n    =  a            a      b              (      a      −      b      )      (      4      a      −      3      b      )      Partitional fractioning in two variables is not unique. Make polynomial partitional fractioning considering a as independent variable            a      b              (      a      −      b      )      (      4      a      −      3      b      )        =      A          a      −      b        +      B          4      a      −      3      b        a  b  =  4  a  A  −  3  b  A  +  B  a  −  B  b  =  (  4  A  +  B  )  a  −  (  3  A  +  B  )  b  4  A  +  B  =  b  ,  3  A  +  B  =  0  ⇒  A  =  b  ,  B  =  −  3  b            a      b              (      a      −      b      )      (      4      a      −      3      b      )        =      b          a      −      b        −            3      b              4      a      −      3      b                  a      b              (      a      −      b      )      (      4      a      −      3      b      )        =      b          a      −      b        +      1    2    −      (                  3        b                    4        a        −        3        b              +          1      2        )  Combining       b          a      −      b        +      1    2   and             3      b              4      a      −      3      b        +      1    2              a      b              (      a      −      b      )      (      4      a      −      3      b      )        =            a      +      b              2      (      a      −      b      )        −            4      a      +      3      b              2      (      4      a      −      3      b      )

Petrovcic2x · Accepted Answer

12      n              (              4        n            −              3        n            )      (              4                  n          +          1                    −              3                  n          +          1                    )        =                    4        n            ⋅              3        n                    (              4        n            −              3        n            )      (      4      ⋅              4        n            −      3      ⋅              3        n            )      Divide top and bottom by       3          2      n       and write   x  =            4      n              3      n       and we get      x          (      x      −      1      )      (      4      x      −      3      )      In partial fractions, this is      1          x      −      1        −      3          4      x      −      3        =      1          x      −      1        −      1                  4        3            x      −      1      So the expression is      1          (              4        3                    )        n            −      1        −      1          (              4        3                    )                  n          +          1                    −      1      or, equivalently,            3      n                      4        n            −              3        n              −            3              n        +        1                            4                  n          +          1                    −              3                  n          +          1

Solve the equation: S n </msub> = 12 ( 4 −<!--

Answered question

Answer & Explanation

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