Giá trị của \((-1)^{n} \mathrm{C}_{n}^{0}+(-1)^{n-1} 2 \mathrm{C}_{n}^{1}+\cdots+(-1)^{n-k} 2^{k} \mathrm{C}_{n}^{k}+\cdots+2^{n} \mathrm{C}_{n}^{n}\) là:
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Lời giải:
Báo sai\(\)\(\text { Ta có }(1-x)^{n}=\mathrm{C}_{n}^{0}-\mathrm{C}_{n}^{1} x+\mathrm{C}_{n}^{2} x^{2}+\cdots+(-1)^{k} \mathrm{C}_{n}^{k} x^{k}+\cdots+(-1)^{n} \mathrm{C}_{n}^{n} x^{n} \text { . }\)
Cho x=2 ta được
\(\begin{array}{r} (1-2)^{n}=\mathrm{C}_{n}^{0}-2 \mathrm{C}_{n}^{1}+2^{2} \mathrm{C}_{n}^{2}-2^{3} \mathrm{C}_{n}^{3}+\cdots+(-1)^{k} 2^{k} \mathrm{C}_{n}^{k}+\cdots+(-1)^{n} 2^{n} \mathrm{C}_{n}^{n} \\ \Leftrightarrow(-1)^{n}=\mathrm{C}_{n}^{0}-2 \mathrm{C}_{n}^{1}+2^{2} \mathrm{C}_{n}^{2}-2^{3} \mathrm{C}_{n}^{3}+\cdots+(-1)^{k} 2^{k} \mathrm{C}_{n}^{k}+\cdots+(-1)^{n} 2^{n} \mathrm{C}_{n}^{n} \end{array}\)
\(\begin{array}{l} \Leftrightarrow(-1)^{n}(-1)^{n}=(-1)^{n} \mathrm{C}_{n}^{0}+(-1)^{-1}(-1)^{n} 2 \mathrm{C}_{n}^{1}+(-1)^{-2}(-1)^{n} 2^{2} \mathrm{C}_{n}^{2}+\cdots+(-1)^{-k}(-1)^{n} 2^{k} \mathrm{C}_{n}^{k}+\cdots +(-1)^{n}(-1)^{n} 2^{n} \mathrm{C}_{n}^{n} \\ \Leftrightarrow 1=(-1)^{n} \mathrm{C}_{n}^{0}+(-1)^{n-1} 2 \mathrm{C}_{n}^{1}+(-1)^{n-2} 2^{2} \mathrm{C}_{n}^{2}+(-1)^{n-3} 2^{3} \mathrm{C}_{n}^{3}+\cdots++(-1)^{n-k} 2^{k} \mathrm{C}_{n}^{k}+\cdots+2^{n} \mathrm{C}_{n}^{n} \end{array}\)
Vậy \((-1)^{n} \mathrm{C}_{n}^{0}+(-1)^{n-1} 2 \mathrm{C}_{n}^{1}+\cdots+(-1)^{n-k} 2^{k} \mathrm{C}_{n}^{k}+\cdots+2^{n} \mathrm{C}_{n}^{n}=1\)
