Question

Giải phương trình :

\(\left|x+\frac{1}{1\times5}\right|+\left|x+\frac{1}{5\times9}\right|+\left|x+\frac{1}{9\times13}\right|+...+\left|x+\frac{1}{397\times401}\right|=101\times x\)

Answer 1

Ta có: \(\left|x+\frac{1}{1.5}\right|+\left|x+\frac{1}{5.9}\right|+\left|x+\frac{1}{9.13}\right|+...+\left|x+\frac{1}{397.401}\right|\ge0\)

\(\Rightarrow101x\ge0\)

\(\Rightarrow x\ge0\)

\(\Rightarrow x+\frac{1}{1.5}+x+\frac{1}{5.9}+...+x+\frac{1}{397.401}=101x\)

\(\Rightarrow100x+\left(\frac{1}{1.5}+\frac{1}{5.9}+...+\frac{1}{397.401}\right)=101x\)

\(\Rightarrow\frac{1}{4}\left(\frac{4}{1.5}+\frac{4}{5.9}+...+\frac{4}{397.401}\right)=x\)

\(\Rightarrow x=\frac{1}{4}\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+...+\frac{1}{397}-\frac{1}{401}\right)\)

\(\Rightarrow x=\frac{1}{4}\left(1-\frac{1}{401}\right)\)

\(\Rightarrow x=\frac{1}{4}.\frac{400}{401}\)

\(\Rightarrow x=\frac{100}{401}\)

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