Try to figure out a function f ( n ) which takes the input n, where n ∈ N , and outputs the sum ∑ i = 0 i = n k j i till the n t h value where k ∈ R &amp; j ∈ N . For example ∑ i = 0 i = n 3 2 i

Question

Try to figure out a function   f  (  n  ) which takes the input n, where   n  ∈      N  , and outputs the sum       ∑          i      =      0              i      =      n                  k              j        i             till the       n          t      h       value where   k  ∈      R   &amp;amp;   j  ∈      N  . For example       ∑          i      =      0              i      =      n                  3              2        i

Jimena Torres · Accepted Answer

Let   a  ∈      R  .  S  (  n  )  =  a      k          0        +  a      k          j        +  a      k          2      j        +  a      k          3      j        +  ⋯  +  a      k          j      n            k          j        S  (  n  )  =  a      k          j        +  a      k          2      j        +  a      k          3      j        +  a      k          4      j        +  ⋯  +  a      k          j      n      +      j      So,  S  (  n  )  −      k          j        S  (  n  )  =  a      k          0        −  a      k          j      n      +      j        S  (  n  )  (  1  −      k          j        )  =  a      k          0        −  a      k          j      n      +      j        S  (  n  )  =            a              k                  0                    −      a              k                  j          n          +          j                            (      1      −              k                  j                    )        S  (  n  )  =            a      (      1      −              k                  j          n          +          j                    )              (      1      −              k                  j                    )

Try to figure out a function f(n) which takes the input n, where n∈N, and outputs the sum ∑i=ni=0kji till the nth value where k∈R & j∈N

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