Tính nhanh biểu thức dưới đây 1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ... + 1/(1+2+3+4+...+50)

\[1 + \frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + \frac{1}{{1 + 2 + 3 + 4}} + ... + \frac{1}{{1 + 2 + 3 + 4 + ... + 50}}\]

\[ = 1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{{10}} + ... + \frac{1}{{1\,\,275}}\]

\[ = \frac{2}{2} + \frac{2}{6} + \frac{2}{{12}} + \frac{2}{{20}} + ... + \frac{2}{{2\,\,550}}\]

\[ = \frac{2}{{1 \cdot 2}} + \frac{2}{{2 \cdot 3}} + \frac{2}{{3 \cdot 4}} + \frac{2}{{4 \cdot 5}} + ... + \frac{2}{{50 \cdot 51}}\]

\[ = 2 \cdot \left( {\frac{1}{{1 \cdot 2}} + \frac{1}{{2 \cdot 3}} + \frac{1}{{3 \cdot 4}} + \frac{1}{{4 \cdot 5}} + ... + \frac{1}{{50 \cdot 51}}} \right)\]

\[ = 2 \cdot \left( {1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{{50}} - \frac{1}{{51}}} \right)\]

\[ = 2 \cdot \left( {1 - \frac{1}{{51}}} \right)\]

\[ = 2 \cdot \frac{{50}}{{51}} = \frac{{100}}{{51}}.\]