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填不同的自然数
1 9 = 1 ( ) + 1 ( ) + 1 ( ) + 1 ( ) + 1 ( ) \frac{1}{9}=\frac{1}{(\text{ })}+\frac{1}{(\text{ })}+\frac{1}{(\text{ })}+\frac{1}{(\text{ })}+\frac{1}{(\text{ })} 91=( )1+( )1+( )1+( )1+( )1
推理:
1 = 1 − 1 2 + 1 2 − 1 3 + 1 3 + … − 1 n + 1 n = 1 − 1 2 + 1 2 − 1 3 + 1 3 − ⋯ − 1 n + 1 n = 1 × 2 1 × 2 − 1 × 1 2 × 1 + 1 × 3 2 × 3 − 1 × 2 3 × 2 + 1 × 4 3 × 4 − ⋯ − 1 × ( n − 1 ) n × ( n − 1 ) + 1 n = 2 − 1 1 × 2 + 3 − 2 2 × 3 + 4 − 3 3 × 4 + ⋯ + n − ( n − 1 ) ( n − 1 ) × n + 1 n = 1 1 × 2 + 1 2 × 3 + 1 3 × 4 + ⋯ + 1 ( n − 1 ) × n + 1 n = 1 1 × 2 + 1 2 × 3 + 1 3 × 4 + ⋯ + 1 ( n − 1 ) × n + 1 n = 1 1 × 2 + ⋯ + 1 ( n − 1 ) × n + 1 n \begin{equation} \begin{split} 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}-\dots -\frac{1}{n}}+\frac{1}{n}\\ &=\boxed{\frac{1\times2}{1\times2}-\frac{1\times1}{2\times1}}+\boxed{\frac{1\times3}{2\times3}-\frac{1\times2}{3\times2}}+\boxed{\frac{1\times4}{3\times4}-\dots -\frac{1\times(n-1)}{n\times(n-1)}}+\frac{1}{n}\\ &=\boxed{\frac{2-1}{1\times2}}+\boxed{\frac{3-2}{2\times3}}+\boxed{\frac{4-3}{3\times4}}+\dots +\boxed{\frac{n-(n-1)}{(n-1)\times n}}+\frac{1}{n}\\ &=\boxed{\frac{1}{1\times2}}+\boxed{\frac{1}{2\times3}}+\boxed{\frac{1}{3\times4}}+\dots +\boxed{\frac{1}{(n-1)\times n}}+\frac{1}{n}\\ &=\boxed{\frac{1}{1\times2}}+\boxed{\frac{1}{2\times3}}+\boxed{\frac{1}{3\times4}}+\dots +\boxed{\frac{1}{(n-1)\times n}}+\frac{1}{n}\\ &=\frac{1}{1\times2}+\dots +\frac{1}{(n-1)\times n}+\frac{1}{n}\\ \end{split} \end{equation} 1=1−21+21−31+31+…−n1+n1=1−21+21−31+31−⋯−n1+n1=1×21×2−2×11×1+2×31×3−3×21×2+3×41×4−⋯−n×(n−1)1×(n−1)+n1=1×22−1+2×33−2+3×44−3+⋯+(n−1)×nn−(n−1)+n1=1×21+2×31+3×41+⋯+(n−1)×n1+n1=1×21+2×31+3×41+⋯+(n−1)×n1+n1=1×21+⋯+(n−1)×n1+n1
将公式 ( 1 ) (1) (1)两边同时乘以 1 K \frac{1}{K} K1得到下列公式:
1 K × 1 = 1 K × ( 1 1 × 2 + ⋯ + 1 ( n − 1 ) × n + 1 n ) = 1 1 × 2 × K + ⋯ + 1 ( n − 1 ) × n × K + 1 n × K \begin{equation} \begin{split} \frac{1}{K} \times 1&=\frac{1}{K} \times (\frac{1}{1\times2}+\dots +\frac{1}{(n-1)\times n}+\frac{1}{n}) \\ &=\frac{1}{1\times2\times K}+\dots +\frac{1}{(n-1)\times n\times K}+\frac{1}{n\times K} \end{split} \end{equation} K1×1=K1×(1×21+⋯+(n−1)×n1+n1)=1×2×K1+⋯+(n−1)×n×K1+n×K1
解
使用万能公式 ( 2 ) (2) (2)解上述题目:
- K = 9 K=9 K=9 分母为9
- n = 5 n=5 n=5 将 1 9 \frac{1}{9} 91拆成 5 5 5项
1 9 = 1 1 × 2 × 9 + 1 2 × 3 × 9 + 1 3 × 4 × 9 + 1 4 × 5 × 9 + 1 5 × 9 = 1 18 + 1 54 + 1 108 + 1 180 + 1 45 \begin{equation} \begin{split} \frac{1}{9}&=\frac{1}{1\times2\times9}+\frac{1}{2\times3\times9}+\frac{1}{3\times4\times9}+\frac{1}{4\times5\times9}+\frac{1}{5\times 9} \\ &= \frac{1}{18}+\frac{1}{54}+\frac{1}{108}+\frac{1}{180}+\frac{1}{45} \\ \end{split} \end{equation} 91=1×2×91+2×3×91+3×4×91+4×5×91+5×91=181+541+1081+1801+451
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