\(S=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+....+\frac{1}{100}\left(1+2+3+....+100\right)\)
\(=1+\frac{1}{2}.\frac{2.3}{2}+\frac{1}{3}.\frac{3.4}{2}+.....+\frac{1}{100}.\frac{100.101}{2}\)
\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+.....+\frac{101}{2}\)
\(=\frac{2+3+4+....+101}{2}\)
\(=\frac{\frac{101.102}{2}-1}{2}\)
\(=2575\)
Vậy \(S=2575\)