\(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)
\(=3.\left(\frac{1}{1.4}-\frac{1}{4.7}\right)+3.\left(\frac{1}{4.7}-\frac{1}{7.10}\right)+...+3.\left(\frac{1}{54.57}-\frac{1}{57.60}\right)\)
\(=3\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...-\frac{1}{54.57}+\frac{1}{54.57}-\frac{1}{57.60}\right)\)
\(=3\left(\frac{1}{1.4}-\frac{1}{57.60}\right)\)
\(=3\left(\frac{1}{4}-\frac{1}{3420}\right)\)
\(=3\left(\frac{855}{3420}-\frac{1}{3420}\right)\)
\(=3.\frac{427}{1710}\)
\(=\frac{427}{570}\)