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\(\frac{5}{1+2+3}+\frac{5}{1+2+3+4}+...+\frac{5}{1+2+3+...+100}\)
\(=\frac{5}{\left(3+1\right).3:2}+\frac{5}{\left(4+1\right).4:2}+...+\frac{5}{\left(100+1\right).100:2}\)
\(=\frac{5}{4.3:2}+\frac{5}{5.4:2}+...+\frac{5}{100.101:2}\)
\(=\frac{10}{3.4}+\frac{10}{4.5}+...+\frac{10}{100.101}\)
\(=10\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{100.101}\right)\)
\(=10\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{100}-\frac{1}{101}\right)\)
\(=10\left(\frac{1}{3}-\frac{1}{101}\right)\)
\(=10.\frac{98}{303}\)
\(=\frac{980}{303}\)