binomial

find the coefficient of x_⁵⁰in the expression
(1+x)_¹⁰⁰⁰+2x(1+x)_⁹⁹⁹+3x_²(1+x)_⁹⁹⁸+...+1000x_⁹⁹⁹(1+x)+1001x¹⁰⁰⁰

may be it is a sum from any common book bu i do not have the solution .i found it in my state book exercise.

3 Answers

gvn

= (1+x)¹⁰⁰⁰.(1+2t+3t²................1001t¹⁰⁰⁰)
=(1+x)¹⁰⁰⁰. f(t)

where t =x/1+x

f(t) = d/dt ( âˆ« f(t) dt )

now proceed

i must accept this is quite a long sum

another way is

Î£ r.C^1001-r_51-r
but simplifying this is not very easy

couldn"t understand integrating it and then deriving it will bring me to the same step

no see itll be diffrentiation of t(1-t¹⁰⁰¹)/1-t

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