\(S=\frac{38}{25}+\frac{9}{10}-\frac{11}{15}+\cdot\cdot\cdot+\frac{197}{4851}-\frac{199}{4950}\)
\(\Rightarrow S=\frac{38}{25}+\frac{18}{20}-\frac{22}{30}+\cdot\cdot\cdot+\frac{394}{9702}-\frac{398}{9900}\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\left(\frac{9}{20}-\frac{11}{30}+\cdot\cdot\cdot+\frac{197}{9702}-\frac{199}{9900}\right)\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\left(\frac{9}{4\cdot5}-\frac{11}{5\cdot6}+\cdot\cdot\cdot+\frac{197}{98\cdot99}-\frac{199}{99\cdot100}\right)\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\left(\frac{1}{4}+\frac{1}{5}-\frac{1}{5}-\frac{1}{6}+\cdot\cdot\cdot-\frac{1}{99}-\frac{1}{100}\right)\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\left(\frac{1}{4}-\frac{1}{100}\right)\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\left(\frac{25}{100}-\frac{1}{100}\right)\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\frac{24}{100}\)
\(\Rightarrow S=\frac{38}{25}+2\cdot\frac{6}{25}\)
\(\Rightarrow S=\frac{38}{25}+\frac{12}{25}\)
\(\Rightarrow S=\frac{50}{25}=2\)
