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Question Number 31229 by abdo imad last updated on 03/Mar/18

$$simplify{A}_n=C_n^1+2C_n^2+3C_n^3+...+n{C}_n^n.$$

Commented by abdo imad last updated on 05/Mar/18

$$letintroducethepolynomialp\left(x\right)=\sum _{k=0}^n{C}_n^k{x}^k$$$$wehavep\left(x\right)={(x+1)}^n\Rightarrow p^{{}^{\prime }}\left(x\right)=n{(x+1)}^{n-1}from$$$$anotherside{p}^{{}^{\prime }}\left(x\right)=\sum _{k=1}^{n}k{C}_n^k{x}^{k-1}andwehave$$$$A_n=p^{{}^{\prime }}\left(1\right)=n{2}^{n-1}.$$$$$$

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