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Question Number 163576 by cortano1 last updated on 08/Jan/22

What is the coefficient of x^(2020) in (1+x+x^2 +x^3 +...+x^(2020) )^(2021)

Whatisthecoefficientofx2020in(1+x+x2+x3+...+x2020)2021

Answered by bobhans last updated on 08/Jan/22

f(x)=(1+x+x^2 +...+x^(2020) )^(2021) f(x)= ((1/(1−x)))^(2021) = Σ_(m=0) ^(2021) (((2021+m−1)),(( m)) ) x^k take k=2020 ⇒coefficient of x^(2020) = (((4040)),((2020)) )

f(x)=(1+x+x2+...+x2020)2021f(x)=(11−x)2021=∑2021m=0(2021+m−1m)xktakek=2020⇒coefficientofx2020=(40402020)

Answered by mr W last updated on 08/Jan/22

since you ask for the coef. of x^(2020) , you can add x^(2021) , x^(2022) , ..., ∞ into the expression. that wont change the coef. of x^(2020) . that means coef. of x^(2020) in (1+x+x^2 +x^3 +...+x^(2020) )^(2021) is the same as coef. of x^(2020) in (1+x+x^2 +x^3 +...+x^(2020) +x^(2021) +...)^(2021) . (1+x+x^2 +x^3 +...+x^(2020) +x^(2021) +...)^(2021) =((1/(1−x)))^(2021) =Σ_(k=0) ^∞ C_(2020) ^(k+2020) x^k the coef. of x^(2020) is then C_(2020) ^(2020+2020) =C_(2020) ^(4040)

sinceyouaskforthecoef.ofx2020,youcanaddx2021,x2022,...,∞intotheexpression.thatwontchangethecoef.ofx2020.thatmeanscoef.ofx2020in(1+x+x2+x3+...+x2020)2021isthesameascoef.ofx2020in(1+x+x2+x3+...+x2020+x2021+...)2021.(1+x+x2+x3+...+x2020+x2021+...)2021=(11−x)2021=∑∞k=0C2020k+2020xkthecoef.ofx2020isthenC20202020+2020=C20204040

Commented by mr W last updated on 08/Jan/22

but if you ask for the coef. of x^(2050) , then you must proceed normally as following. (1+x+x^2 +x^3 +...+x^(2020) )^(2021) =[((1−x^(2021) )/(1−x))]^(2021) =(1−x^(2021) )^(2021) Σ_(k=0) ^∞ C_k ^(k+2020) x^k coef. of x^(2020) is C_(2020) ^(2020+2020) coef. of x^(2050) is C_(2050) ^(2050+2020) −2021C_(29) ^(29+2020)

butifyouaskforthecoef.ofx2050,thenyoumustproceednormallyasfollowing.(1+x+x2+x3+...+x2020)2021=[1−x20211−x]2021=(1−x2021)2021∑∞k=0Ckk+2020xkcoef.ofx2020isC20202020+2020coef.ofx2050isC20502050+2020−2021C2929+2020

Commented by Tawa11 last updated on 08/Jan/22

Great sir

Greatsir

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