Update 文章 “使用归纳法证明二项式定理”
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2026-04-16 21:22:20 +08:00
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@@ -22,14 +22,14 @@ $$
试图使左右两边相等,需要使得左边分母为$k!(n+1-k)!$,所以通分得:
$$
\begin{align}
\begin{align*}
& \frac{n!}{(k-1)!(n-k+1)!} + \frac{n!}{k!(n-k)!} \\\\
&= \frac{n!k}{k!(n-k+1)!} + \frac{n!(n-k+1)}{k!(n-k+1)!} \\\\
&= \frac{n!k+n!(n-k+1)}{k!(n-k+1)!} \\\\
&= \frac{n!(n+1)}{k!(n-k+1)!} \\\\
&= \frac{(n+1)!}{k!(n+1-k)!} \\\\
&= \binom{n+1}{k}. \tag*{$\blacksquare$}
\end{align}
\end{align*}
$$
随后证明二项式定理