\(M=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{10.11}-\frac{1}{11.12}\)

\(=\frac{1}{2}-\frac{1}{11.12}\)

 \(=\frac{65}{132}\)

65/132

65/132

M = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ... + 1/10.11.12

M = 1/2.(2/1.2.3 + 2/2.3.4 + 2/3.4.5 + ... + 2/10.11.12)

M = 1/2.(1/1.2 - 1/2.3 + 1/2.3- 1/3.4 + 1/3.4 - 1/4.5 + ... + 1/10.11 - 1/11.12)

M = 1/2.(1/1.2 - 1/11.12)

M = 1/4 - 1/264

M = 65/264

M=65/264.

Kết quả 65/264

   -chúc các bạn học tốt-

\(M=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{10.11}-\frac{1}{11.12}\right)\)

\(M=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{11.12}\right)\)

\(M=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{132}\right)\)

\(D=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+...+\frac{1}{10\cdot11\cdot12}\)

\(D=\frac{1}{2}\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{10\cdot11\cdot12}\right)\)

\(D=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{10\cdot11}-\frac{1}{11\cdot12}\right)\)

\(D=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{132}\right)=...\)

\(D=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{10.11.12}\)

\(D=\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{10.11.12}\right).\frac{1}{2}\)

\(D=\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{10.11}-\frac{1}{11.12}\right).\frac{1}{2}\)

\(D=\left(\frac{1}{1.2}-\frac{1}{11.12}\right).\frac{1}{2}\)

\(D=\frac{65}{132}.\frac{1}{2}\)

\(D=\frac{65}{264}\)

\(2M=2\cdot\left(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+.....+\frac{1}{10\cdot11\cdot12}\right)\)

\(2M=\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+.....+\frac{2}{10\cdot11\cdot12}\)

\(2M=\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+.....+\frac{1}{10\cdot11}-\frac{1}{11\cdot12}\)

\(2M=\frac{1}{1\cdot2}-\frac{1}{11\cdot12}\)

\(2M=\frac{1}{2}-\frac{1}{132}\)

\(2M=\frac{66}{132}-\frac{1}{132}\)

\(2M=\frac{65}{132}\)

\(M=\frac{65}{132}:2\)

\(M=\frac{65}{264}\)

2A=2(1/1.2.3+1/2.3.4+...+1/98.99.100)

2A=1/1.2-1/2.3+1/2.3-1/3.4+1/3.4-...+1/98.99-1/99.100

2A=1/1.2-1/99.100

2A=4949/9900

A=4949/9900:2

A=4949/19800

                                         Vậy A=4949/198000

Lưu Trường An làm đúng rồi

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2010.2011.2012}\)

\(\Rightarrow\frac{1}{4}A=\frac{1}{1.2.3.4}+\frac{1}{2.3.4\left(5-1\right)}+\frac{1}{3.4.5\left(6-2\right)}+...+\frac{1}{2010.2011.2012.\left(2013-2009\right)}\)

\(\Rightarrow\frac{1}{4}A=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}-\frac{1}{1.2.3.4}+\frac{1}{3.4.5.6}-\frac{1}{2.3.4.5}+...+\frac{1}{2010.2011.2012.2013}-\frac{1}{2009.2010.2011.2012}\)

\(\Rightarrow\frac{1}{4}A=\frac{1}{2010.2011.2012.2013}\)

\(\Rightarrow A=\frac{4}{2010.2011.2012.2013}\)

\(\Rightarrow A=\frac{1}{2010.2011.503.2013}\)

Ta có : \(B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

\(\Leftrightarrow2B=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{18.19}-\frac{1}{19.20}\)

\(=\frac{1}{2}-\frac{1}{19.20}=\frac{189}{380}\)

\(\Rightarrow B=\frac{189}{760}\)

\(B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{18.19.20}\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\frac{1}{2}\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\frac{1}{2}\left(\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{19.20}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{380}\right)\)

\(=\frac{1}{2}.\frac{189}{380}=\frac{189}{760}\)

\(B=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\right).\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right).\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)\)

\(=\frac{179}{760}\)

M= 1 - 1/2 + 1/2 - 1/3 +1/3 -1 /4 + 1/4 - 1/5 + .....+ 1/10 - 1/11 + 1/11 -1/12​

   = 1 - 1/12

   = 11/12