Cho biểu thức M = 1/(1+2+3) + 1/(1+2+3+4) +... + 1/(1+2+3+4+..+59). Chứng minh rằng: M < 2/3

Ta có:

\[\frac{1}{{1 + 2 + 3}} + \frac{1}{{1 + 2 + 3 + 4}} + ... + \frac{1}{{1 + 2 + 3 + 4 + ... + 59}}\]

\[ = \frac{1}{{\frac{{3 \cdot 4}}{2}}} + \frac{1}{{\frac{{4 \cdot 5}}{2}}} + ... + \frac{1}{{\frac{{59 \cdot 60}}{2}}}\]

\[ = \frac{2}{{3 \cdot 4}} + \frac{2}{{4 \cdot 5}} + ... + \frac{2}{{59 \cdot 60}}\]

\[ = 2 \cdot \left( {\frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{{59}} - \frac{1}{{60}}} \right)\]

\[ = 2 \cdot \left( {\frac{1}{3} - \frac{1}{{60}}} \right)\]

\[ = 2 \cdot \frac{{19}}{{60}}\]

\[ = \frac{{38}}{{60}}\]

Mà \[\frac{{38}}{{60}} < \frac{{40}}{{60}} = \frac{2}{3}\]

Vậy \[M < \frac{2}{3}.\]