e,\(A=\frac{1}{2}+\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}=\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{6}\right)+\left(1-\frac{1}{12}\right)+\left(1-\frac{1}{20}\right)+\left(1-\frac{1}{20}\right)+\left(1-\frac{1}{42}\right)\)

\(\Rightarrow A=1-\frac{1}{2}+1-\frac{1}{6}+1-\frac{1}{12}+1-\frac{1}{20}+1-\frac{1}{30}+1-\frac{1}{42}=4-\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}\right)\)

\(\Rightarrow A=4-\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}\right)=4-\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}\right)\)

\(\Rightarrow A=4-\left(\frac{1}{1}-\frac{1}{7}\right)=4-\frac{6}{7}=3\frac{1}{7}\)

Nohara shin nosuke mình xin giúp bạn trả lời câu hỏi:

1/2 +1/4 +1/6 +......+1/100

= 1/2 + 1/2*(1/2+1/3+...+1/50)

1/2 + 1/3 = (2+3)/(2.3) 
tiếp với 3 số 
1/2 + 1/3 + 1/4 =(2.3+3.4+4.2)/(2.3.4) 
với 4 số: 
1/2+1/3+1/4+1/5 = (2.3.4+3.4.5+4.5.2+5.2.3)/(2.3.4.5) 

Với N-1 số 
1/2 + 1/3+1/4+...+1/N= (2.3.4...(N-1) + 3.4.5....N + 4.5...2 + ...) /(2.3.4.....N) 

bạn cứ áp dụng quy luật mà làm nha