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simplify-A-n-C-n-1-2-C-n-2-3-C-n-3-n-C-n-n-

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Question Number 31229 by abdo imad last updated on 03/Mar/18
simplify A_n = C_n ^1 +2 C_n ^2 +3 C_n ^3 +... + n C_n ^n .
simplifyAn=Cn1+2Cn2+3Cn3+…+nCnn.
Commented by abdo imad last updated on 05/Mar/18
let introduce the polynomial p(x)=Σ_(k=0) ^n C_n ^k x^k we have p(x)= (x+1)^n ⇒p^′ (x)=n(x+1)^(n−1) from another side p^′ (x)= Σ_(k=1) ^n k C_n ^k x^(k−1) and we have A_n =p^′ (1)= n 2^(n−1) .
letintroducethepolynomialp(x)=∑k=0nCnkxkwehavep(x)=(x+1)n⇒p′(x)=n(x+1)n−1fromanothersidep′(x)=∑k=1nkCnkxk−1andwehaveAn=p′(1)=n2n−1.

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