Find the coefficient of x n in ( x 2 + x 3 + x 4 + ⋯ ) 5 .I have got stuck on this question, though I realise that I have probably got really close to an answer. This is how I approached it: f ( x ) = ( x 2 + x 3 + x 4 + ⋯ ) 5 = x 10 ( 1 + x + x 2 + ⋯ ) 5 = x 10 ( 1 − x m 1 − x ) 5 = x 10 ( 1 − x m ) 5 ( 1 − x ) − 5 . Then I have used the binomial theorem: ( 1 − x m ) 5 = ∑ i = 0 5 ( 5 i ) ( − 1 ) i ( x m ) i , ( 1 − x ) − 5 = ∑ j = 0 ∞ ( 4 + j j ) ( x ) j . Therefore, f ( x ) = x 10 ⋅ ( ∑ i = 0 5 ( 5 i ) ( − 1 ) i ( x m ) i ) ⋅ ( ∑ j = 0 ∞ ( 4 + j j ) x j ) ,and as I work out coefficient of x n I arrive to: [ x n ] = ( 5 0 ) ( n − 6 n − 10 ) I think my answer is incomplete and unfortunately, I don't have a solution to check it. I would really appreciate your help. Thank you

Question

Find the coefficient of       x    n   in   (      x    2    +      x    3    +      x    4    +  ⋯      )    5  .I have got stuck on this question, though I realise that I have probably got really close to an answer. This is how I approached it:                    f        (        x        )                            =        (                  x          2                +                  x          3                +                  x          4                +        ⋯                  )          5                                                  =                  x                      10                          (        1        +        x        +                  x          2                +        ⋯                  )          5                                                  =                  x                      10                                                (                                          1                −                                  x                                      m                                                                              1                −                x                                      )                    5                                                  =                  x                      10                          (        1        −                  x                      m                                    )          5                (        1        −        x                  )                      −            5                          .            Then I have used the binomial theorem:                    (        1        −                  x                      m                                    )          5                                    =                  ∑                      i            =            0                    5                                                    (                                            5            i                                              )                                      (        −        1                  )          i                (                  x          m                          )          i                ,                            (        1        −        x                  )                      −            5                                              =                  ∑                      j            =            0                    ∞                                                    (                                                          4              +              j                        j                                              )                                      (        x                  )          j                .            Therefore,   f  (  x  )  =      x          10        ⋅      (          ∑              i        =        0            5                                (                            5        i                              )                      (    −    1          )      i        (          x      m              )      i        )    ⋅      (          ∑              j        =        0            ∞                                (                                      4          +          j                j                              )                            x      j        )    ,and as I work out coefficient of       x    n   I arrive to:  [      x    n    ]  =                    (                    5      0                      )                                (                            n        −        6                    n        −        10                            )            I think my answer is incomplete and unfortunately, I don&#039;t have a solution to check it. I would really appreciate your help. Thank you

jmibanezla · Accepted Answer

Step 1As you had before, we have  f  (  x  )  =  (      x    2    +      x    3    +  …      )    5    f  (  x  )  =      x          10        (  1  +  x  +  …      )    5    f  (  x  )  =            x              10                    (      1      −      x              )        5              f  (  x  )  =            x              10                    (      1      −      x              )        5              f  (  x  )  =      x          10            ∑          k      =      0        ∞                      (                            k        +        4            4                      )                  x    k    f  (  x  )  =      ∑          k      =      0        ∞                      (                            k        +        4            4                      )                  x          k      +      10      Step 2Hence, the coefficient of       x    n   is                     (                            n        −        6            4                      )            .