Bài giải:

$A=\left( 1-\frac{1}{1+2} \right)\times \left( 1-\frac{1}{1+2+3} \right)\times \left( 1-\frac{1}{1+2+3+4} \right)\times ...\times \left( 1-\frac{1}{1+2+3+...+2018} \right)$

$=\left( 1-\frac{1}{3} \right)\times \left( 1-\frac{1}{6} \right)\times \left( 1-\frac{1}{10} \right)\times ...\times \left( 1-\frac{1}{2037171} \right)$

$=\frac{2}{3}\times \frac{5}{6}\times \frac{9}{10}\times ...\times \frac{2037170}{2037171}$

$=\frac{4}{6}\times \frac{10}{12}\times \frac{18}{20}\times ...\times \frac{4074340}{4074342}$

$=\frac{1\times 4}{2\times 3}\times \frac{2\times 5}{3\times 4}\times \frac{3\times 6}{4\times 5}\times ...\times \frac{2017\times 2020}{2018\times 2019}$

$=\frac{\left( 1\times 2\times 3\times ...\times 2017 \right)\times \left( 4\times 5\times 6\times ...2020 \right)}{\left( 2\times 3\times 4\times ...\times 2018 \right)\times \left( 3\times 4\times 5\times ...\times 2019 \right)}$

$=\frac{1}{2018}\times \frac{2020}{3}=\frac{1010}{3027}$

Chú ý: 4074342 = 2018 x 2019 (vì 1 + 2 + 3 + … + 2018 = (2018 + 1) x 2018 : 2 = 2019 x 2018 : 2 = 4074342 : 2)