$A=\left( 1-\frac{1}{1+2} \right)\times \left( 1-\frac{1}{1+2+3} \right)\times \left( 1-\frac{1}{1+2+3+4} \right)\times ...\times \left( 1-\frac{1}{1+2+3+...+100} \right)$

$=\left( 1-\frac{1}{3} \right)\times \left( 1-\frac{1}{6} \right)\times \left( 1-\frac{1}{10} \right)\times ...\times \left( 1-\frac{1}{5050} \right)$

$=\frac{2}{3}\times \frac{5}{6}\times \frac{9}{10}\times ...\times \frac{5049}{5050}$

$=\frac{4}{6}\times \frac{10}{12}\times \frac{18}{20}\times ...\times \frac{10098}{10100}$

$=\frac{1\times 4}{2\times 3}\times \frac{2\times 5}{3\times 4}\times \frac{3\times 6}{4\times 5}\times ...\times \frac{99\times 102}{100\times 101}$

$=\frac{\left( 1\times 2\times 3\times ...\times 99 \right)\times \left( 4\times 5\times 6\times ...\times 102 \right)}{\left( 2\times 3\times 4\times ...\times 100 \right)\times \left( 3\times 4\times 5\times ...\times 101 \right)}$

$=\frac{1}{100}\times \frac{102}{3}=\frac{17}{50}$