Update 文章 “使用归纳法证明二项式定理”
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@@ -22,14 +22,14 @@ $$
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试图使左右两边相等,需要使得左边分母为$k!(n+1-k)!$,所以通分得:
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$$
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\begin{align}
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\begin{align*}
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& \frac{n!}{(k-1)!(n-k+1)!} + \frac{n!}{k!(n-k)!} \\\\
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&= \frac{n!k}{k!(n-k+1)!} + \frac{n!(n-k+1)}{k!(n-k+1)!} \\\\
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&= \frac{n!k+n!(n-k+1)}{k!(n-k+1)!} \\\\
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&= \frac{n!(n+1)}{k!(n-k+1)!} \\\\
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&= \frac{(n+1)!}{k!(n+1-k)!} \\\\
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&= \binom{n+1}{k}. \tag*{$\blacksquare$}
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\end{align}
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\end{align*}
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$$
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随后证明二项式定理
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