Tính nhanh biểu thức: 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + ... + 1/300
Giải thích
\[\frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + \frac{1}{{15}} + \frac{1}{{21}} + ... + \frac{1}{{300}}\]
\[ = \frac{2}{6} + \frac{2}{{12}} + \frac{2}{{20}} + ... + \frac{2}{{600}}\]
\[ = 2\left( {\frac{1}{{2 \cdot 3}} + \frac{1}{{3 \cdot 4}} + \frac{1}{{4 \cdot 5}} + ... + \frac{1}{{24 \cdot 25}}} \right)\]
\[ = 2\left( {\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{{24}} - \frac{1}{{25}}} \right)\]
\[ = 2 \cdot \left( {\frac{1}{2} - \frac{1}{{25}}} \right)\]
\[ = 2 \cdot \frac{{23}}{{50}} = \frac{{23}}{{25}}.\]