\(\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}\)
\(=\left(1-\frac{1}{6}\right)+\left(1-\frac{1}{12}\right)+\left(1-\frac{1}{30}\right)+\left(1-\frac{1}{42}\right)\)
\(=\left(1+1+1+1\right)-\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}\right)\)
\(=4-\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}\right)\)
\(=4-\left[\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)+\left(\frac{1}{6}-\frac{1}{7}\right)\right]\)
\(=4-\left(\frac{1}{2}-\frac{1}{7}\right)\)
\(=4-\frac{5}{14}\)
\(=\frac{51}{14}\)
\(\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}\)
\(=>\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}=\frac{350}{420}+\frac{385}{420}+\frac{399}{420}+\frac{406}{420}+\frac{410}{420}\)
\(=>\frac{350}{420}+\frac{385}{420}+\frac{399}{420}+\frac{406}{420}+\frac{410}{420}=\frac{350+385+399+406+410}{420}\)
\(=>\frac{350+385+399+406+410}{420}=\frac{1950}{420}=\frac{65}{14}\)
