\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+...+\frac{1}{420}\)

\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{20.21}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{20}-\frac{1}{21}\)

\(=1-\frac{1}{21}\)

\(=\frac{20}{21}\)

\(\left(x+\dfrac{1}{2}\right)+\left(x+\dfrac{1}{6}\right)+\left(x+\dfrac{1}{12}\right)+...+\left(x+\dfrac{1}{420}\right)=20\)

\(\left(x+x+x+...+x\right)+\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{420}\right)=20\) (20 số x)

\(20x+\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{20}-\dfrac{1}{21}\right)=20\)

\(20x+\left(1-\dfrac{1}{21}\right)=20\)

\(20x+\dfrac{20}{21}=20\)

\(20x=20-\dfrac{20}{21}\)

\(20x=\dfrac{400}{21}\)

\(x=\dfrac{400}{21}:20\)

\(x=\dfrac{400}{21}.\dfrac{1}{20}\)

\(x=\dfrac{20}{21}\)

\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+......+\frac{1}{420}+\frac{1}{462}\)

\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{20.21}+\frac{1}{21.22}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+......+\frac{1}{20}-\frac{1}{21}+\frac{1}{21}-\frac{1}{22}\)

\(=1-\frac{1}{22}\)

\(=\frac{22}{22}-\frac{1}{22}=\frac{21}{22}\)

  ta có      1/2+1/6+1/12+1/20+....+1/420+1/462

            =1/1*2+1/2*3+1/3*4+.......+1/20*21+1/21*22

      =1/1-1/2+1/2-1/3+1/3-1/4+.....+1/20-1/21+1/21-1/22

      =1-1/22=21/22     

\(A=\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{420}\\ \Rightarrow A=\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{20\times21}\\ \Rightarrow A=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{20}-\dfrac{1}{21}\\\Rightarrow A=\dfrac{1}{2}-\dfrac{1}{20}=\dfrac{9}{20}\)

Giúp me với

Ta có: \(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{420}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{20}-\dfrac{1}{21}\)

\(=1-\dfrac{1}{21}=\dfrac{20}{21}\)

\(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{420}\\ =\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{20.21}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-....-\dfrac{1}{21}+\dfrac{1}{21}\\ =1-\dfrac{1}{21}\\ =\dfrac{20}{21}\)

\(A=1\frac12+1\frac16+1\frac{1}{12}+\cdots+1\frac{1}{380}+1\frac{1}{420}\)

\(=\left(1+\frac12\right)+\left(1+\frac16\right)+\cdots+\left(1+\frac{1}{420}\right)\)

\(=\left(1+\frac{1}{1\times2}\right)+\left(1+\frac{1}{2\times3}\right)+\cdots+\left(1+\frac{1}{20\times21}\right)\)

\(=20+\frac{1}{1\times2}+\frac{1}{2\times3}+\cdots+\frac{1}{20\times21}\)

\(=20+1-\frac12+\frac12-\frac13+\ldots+\frac{1}{20}-\frac{1}{21}\)

\(=20+1-\frac{1}{21}=21-\frac{1}{21}=\frac{440}{21}\)