Gọi tổng đó là tổng S

Ta có: S = 1/6+1/12+1/30+1/42+1/56+1/72+1/90

=>      S = 1/2.3+1/3.4+1/4.5+1/5.6+1/6.7+1/7.8+1/8.9+1/9.10

=>      S = 1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10

=>      S = 1/2-1/10

=>      S = 5/10-1/10

=>      S=4/10

=>      S=2/5 

Phân số là \(\frac{2}{5}\)

ta thấy: 1/6=1/2x3; 1/12=1/3x4;1/20=1/4x5;.............;1/90=9x10

Đặt A= 1/6+1/12+1/20+...+1/90

A=1/2x3+1/3x4+1/4x5+.....+1/9x10

A=1/2 -1/3+1/3 -1/4+1/4 -1/5+.......+1/9 -1/10

A=1/2-(1/3+1/3)-(1/4+1/4)-......-(1/9+1/9)-1/10

A=1/2-0-0-0-.....0-1/10

A=1/2-1/10

A=2/5

Có công thức \(\dfrac{x}{a\left(a+x\right)}=\dfrac{1}{a}-\dfrac{1}{a+x}\) nhé!

Ví dụ: \(\dfrac{2}{2.4}=\dfrac{1}{2}-\dfrac{1}{4}\)

\(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+\dfrac{1}{56}\)

\(=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}+\dfrac{1}{7.8}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{7}-\dfrac{1}{8}\)

\(=1-\dfrac{1}{8}=\dfrac{7}{8}\)

Dấu . tức là nhân nhé!

\(B=\frac{6}{1\cdot3}+\frac{6}{3\cdot5}+\cdot\cdot\cdot+\frac{6}{97\cdot99}\)

\(\Rightarrow B=3\cdot\left(\frac{2}{1\cdot3}+\cdot\cdot\cdot+\frac{2}{97\cdot99}\right)\)

\(\Rightarrow B=3\cdot\left(1-\frac{1}{3}+\cdot\cdot\cdot+\frac{1}{97}-\frac{1}{99}\right)\)

\(\Rightarrow B=3\cdot\left(1-\frac{1}{99}\right)\)

\(\Rightarrow B=3\cdot\frac{98}{99}\)

\(\Rightarrow B=\frac{98}{33}\)

\(A=\frac{1}{2}+\frac{1}{6}+\cdot\cdot\cdot+\frac{1}{42}\)

\(\Rightarrow A=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdot\cdot\cdot+\frac{1}{6\cdot7}\)

\(\Rightarrow A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\cdot\cdot\cdot+\frac{1}{6}-\frac{1}{7}\)

\(\Rightarrow A=1-\frac{1}{7}\)

\(\Rightarrow A=\frac{6}{7}\)

\(D=\frac{1}{2}+\frac{1}{4}+\cdot\cdot\cdot+\frac{1}{128}\)

\(\Rightarrow2D=1+\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{64}\)

\(\Rightarrow2D-D=\left(1+\cdot\cdot\cdot+\frac{1}{64}\right)-\left(\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{128}\right)\)

\(\Rightarrow D=1-\frac{1}{128}\)

\(\Rightarrow D=\frac{127}{128}\)

\(B=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}\)

\(B=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+\dfrac{1}{6\cdot7}\)

\(B=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\)

\(B=1-\dfrac{1}{7}\)

\(B=\dfrac{6}{7}\)

Quy đồng ik em

\(B=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}\)

\(=\dfrac{6}{7}\)

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

1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 
= 1/2x3 + 1/3x4 + 1/4x5 + 1/5x6 + 1/6x7 + 1/7X8 
=1/2 - 1/3 + 1/3 -1/4 + 1/4 - 1/5 + 1/5 -1/6 + 1/6 - 1/7 + 1/7 - 1/8 
= 1/2 - 1/8 
= 4/8 - 1/8 
= 3/8 

S= \(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}\)

S= 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 +  1/5 -1/6 +1/6 - 1/7 + 1/7 - 1/8

S= 1/2 - 1/ 8 

S= 3/8

S= 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8

  = 1/2 - 1/3 + 1/3 - ...+ 1/7 - 1/8

  = 1/2 - 1/8

  = 3/8

S=(1/2.3) +(1/3.4) + (1/4.5) +...+1/7.8

S=1/2-1/3+1/3-1/4 +1/4-1/5+...+1/7-1/8

S=1/2-1/8

S=3/8