Tính biểu thức sau: A = 1 + 1/3 + 1/6 + 1/10 + 1/15 + ... + 1/120
Giải thích
\[A = 1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + \frac{1}{{15}} + ... + \frac{1}{{120}}\]
\[ = \frac{2}{2} \cdot \left( {\frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + \frac{1}{{15}} + ... + \frac{1}{{120}}} \right)\]
\[ = 2\left( {\frac{1}{6} + \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} + ... + \frac{1}{{240}}} \right)\]
\[ = 2\left( {\frac{1}{{2 \cdot 3}} + \frac{1}{{3 \cdot 4}} + \frac{1}{{4 \cdot 5}} + \frac{1}{{5 \cdot 6}} + ... + \frac{1}{{15 \cdot 16}}} \right)\]
\[ = 2 \cdot \left( {\frac{1}{2} - \frac{1}{{16}}} \right)\]
\[ = 2 \cdot \frac{7}{{16}} = \frac{7}{8}.\]