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- Hướng dẫn giải

$$\begin{gathered} \frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\mathrm{...}+\frac{1}{100.103} \\ =\frac{1}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+\mathrm{...}+\frac{3}{100.103}\right) \\ =\frac{1}{3}\left(\frac{1}{1}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+\mathrm{...}+\frac{1}{100}-\frac{1}{103}\right) \\ =\frac{1}{3}.\left(1-\frac{1}{103}\right)=\frac{34}{103} \end{gathered}$$