填不同的自然数

$$\frac{1}{9}=\frac{1}{()}+\frac{1}{()}+\frac{1}{()}+\frac{1}{()}+\frac{1}{()}$$

推理：

$$\begin{array}{c}\begin{array}{rl}1& =1\boxed{-\frac{1}{2}+\frac{1}{2}}\boxed{-\frac{1}{3}+\frac{1}{3}}+\dots \boxed{-\frac{1}{n}+\frac{1}{n}}\\ & =\boxed{1-\frac{1}{2}}+\boxed{\frac{1}{2}-\frac{1}{3}}+\boxed{\frac{1}{3}-\cdots -\frac{1}{n}}+\frac{1}{n}\\ & =\boxed{\frac{1\times 2}{1\times 2}-\frac{1\times 1}{2\times 1}}+\boxed{\frac{1\times 3}{2\times 3}-\frac{1\times 2}{3\times 2}}+\boxed{\frac{1\times 4}{3\times 4}-\cdots -\frac{1\times (n-1)}{n\times (n-1)}}+\frac{1}{n}\\ & =\boxed{\frac{2-1}{1\times 2}}+\boxed{\frac{3-2}{2\times 3}}+\boxed{\frac{4-3}{3\times 4}}+\cdots +\boxed{\frac{n-(n-1)}{(n-1)\times n}}+\frac{1}{n}\\ & =\boxed{\frac{1}{1\times 2}}+\boxed{\frac{1}{2\times 3}}+\boxed{\frac{1}{3\times 4}}+\cdots +\boxed{\frac{1}{(n-1)\times n}}+\frac{1}{n}\\ & =\boxed{\frac{1}{1\times 2}}+\boxed{\frac{1}{2\times 3}}+\boxed{\frac{1}{3\times 4}}+\cdots +\boxed{\frac{1}{(n-1)\times n}}+\frac{1}{n}\\ & =\frac{1}{1\times 2}+\cdots +\frac{1}{(n-1)\times n}+\frac{1}{n}\end{array}\end{array}$$
将公式$(1)$两边同时乘以$\frac{1}{K}$得到下列公式：
$$\begin{array}{c}\begin{array}{rl}\frac{1}{K}\times 1& =\frac{1}{K}\times (\frac{1}{1\times 2}+\cdots +\frac{1}{(n-1)\times n}+\frac{1}{n})\\ & =\frac{1}{1\times 2\times K}+\cdots +\frac{1}{(n-1)\times n\times K}+\frac{1}{n\times K}\end{array}\end{array}$$

解

使用万能公式$(2)$解上述题目：

• $K=9$ 分母为9
• $n=5$ 将$\frac{1}{9}$拆成$5$项
  $$\begin{array}{c}\begin{array}{rl}\frac{1}{9}& =\frac{1}{1\times 2\times 9}+\frac{1}{2\times 3\times 9}+\frac{1}{3\times 4\times 9}+\frac{1}{4\times 5\times 9}+\frac{1}{5\times 9}\\ & =\frac{1}{18}+\frac{1}{54}+\frac{1}{108}+\frac{1}{180}+\frac{1}{45}\end{array}\end{array}$$