2S=2/1.2.3+2/2.3.4+...+2/8.9.10

2S=1/1.2+1/2.3+1/3.4+1/4.5+....+1/8.9+1/9.10

2S=1-1/2+1/2-1/3+1/3-1/4+...+1/9-1/10

2S=1-1/10

2S=9/10

S=9/10:2

S=9/10.2

S=9/20

1/1.2.3+1/2.3.4+...+1/2007.2008.2009=1-1/2-1/3+1/2-1/3-1/4+...-1/2008-1/2009=1-1/2009=2008/2009

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+.....+\frac{1}{2007.2008.2009}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-.....-\frac{1}{2008.2009}\)

\(=\frac{1}{1.2}-\frac{1}{2008.2009}=\frac{1}{2}-\frac{1}{4034072}=\frac{2017035}{4034072}\)

giúp mình vs ạ

a) \(A=1\cdot2\cdot3-2\cdot3\cdot4+3\cdot4\cdot5+...+98\cdot99\cdot100\)

\(\Rightarrow4A=4\cdot\left(1\cdot2\cdot3+2\cdot3\cdot4+...+98\cdot99\cdot100\right)\)

\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot4+3\cdot4\cdot5\cdot4+...+98\cdot99\cdot100\cdot4\)

\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot\left(5-1\right)+3\cdot4\cdot5\cdot\left(6-2\right)+....+98\cdot99\cdot100\cdot\left(101-97\right)\)

\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot5-1\cdot2\cdot3\cdot4+3\cdot4\cdot5\cdot6-....-97\cdot98\cdot99\cdot100\)

\(\Rightarrow4A=\left(1\cdot2\cdot3\cdot4-1\cdot2\cdot3\cdot4\right)+\left(2\cdot3\cdot4\cdot5-2\cdot3\cdot4\cdot5\right)+...+98\cdot99\cdot100\cdot101\)

\(\Rightarrow4A=0+0+0+...+98\cdot99\cdot100\cdot101\)

\(\Rightarrow4A=98\cdot99\cdot100\cdot101\)

\(\Rightarrow A=\dfrac{98\cdot99\cdot100\cdot101}{4}\)

b) \(B=1+3^2+3^3+3^4+...+3^{99}\)

\(\Rightarrow3B=3\cdot\left(1+3^2+3^3+...+3^{99}\right)\)

\(\Rightarrow3B=3+3^3+3^4+3^5+...+3^{99}+3^{100}\)

\(\Rightarrow3B-B=\left(3+3^3+3^4+...+3^{100}\right)-\left(1+3^2+3^3+...+3^{99}\right)\)

\(\Rightarrow2B=\left(3^3-3^3\right)+\left(3^4-3^4\right)+...+\left(3-1\right)+\left(3^{100}-3^2\right)\)

\(\Rightarrow2B=2+3^{100}-3^2\)

\(\Rightarrow B=\dfrac{2+3^{100}-3^2}{2}\)

bai nay mik lam sai roi nha

      \(\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\right).1428+185.8\)

\(=\frac{2}{2}.\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{37.38.39}\right).1428+185.8\)

\(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{37.38.39}\right).1428+1480\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{37.38}-\frac{1}{38.39}\right).1428+1480\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{38.39}\right).1428+1480\)

\(\frac{1}{2}.\frac{370}{741}.1428+1480\)

\(=\frac{185}{741}.1428+1480\)

\(=356,52+1480=1836,52\)

chỗ\(\frac{185}{741}.1428\)mk làm tròn số lun á nha

mk ko chắc tính đúng hay sai nha nhưng cách làm thì kiểu vậy

\(\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{37.38}-\frac{1}{38.39}\right).1428+185.8\)

\(=\left(\frac{1}{2}-\frac{1}{1482}\right).1482+1840\)

\(=\frac{740}{1482}.1482+1840\)

\(=740+1840\)

\(=2580\)

Ta có: \(\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{8.9.10}\right)x=\frac{23}{45}\)

\(\Rightarrow\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{8.9}-\frac{1}{9.10}\right)x=\frac{23}{45}\)

\(\Rightarrow\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{9.10}\right)x=\frac{23}{45}\)

\(\Rightarrow\frac{1}{2}\left(\frac{1}{2}-\frac{1}{90}\right)x=\frac{23}{45}\)

\(\Rightarrow\frac{11}{45}x=\frac{23}{45}\)

\(\Rightarrow x=\frac{23}{45}:\frac{11}{45}\)

\(\Rightarrow x=\frac{23}{11}\)

Đặt A=\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{8.9.10}\)

    2A=\(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{8.9.10}\)

    2A=\(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}\) \(+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{8.9}-\frac{1}{9.10}\)

    2A=\(\frac{1}{1.2}-\frac{1}{9.10}\)

    2A=\(\frac{22}{45}\)

      A=\(\frac{22}{45}\div2\)

      A=\(\frac{11}{45}\)

\(\Rightarrow\frac{11}{45}.x=\frac{23}{45}\)

             \(x=\frac{23}{45}\div\frac{11}{45}=\frac{23}{11}\)

Vậy x=\(\frac{23}{11}\)

\(\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+....+\frac{1}{8.9.10}\right).x=\frac{23}{45}\)

\(\Leftrightarrow\left[\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{8.9}-\frac{1}{9.10}\right)\right]x=\frac{23}{45}\)

\(\Leftrightarrow\left[\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{9.10}\right)\right].x=\frac{23}{45}\)

\(\Leftrightarrow\left(\frac{1}{2}.\frac{44}{90}\right).x=\frac{23}{45}\)

\(\Leftrightarrow\frac{11}{45}.x=\frac{23}{45}\Rightarrow x=\frac{23}{45}:\frac{11}{45}=\frac{23}{11}\)

nhung sao banj khong phan h ra ro rang,chang nhe den do khong phan h  duoc sao

Ahihi

= 1/2*(1/1*2 - 1/2*3 + 1/2*3 - 1/3*4 + ... + 1/8*9 - 1/9*10) = 1/2*(1/1*2 - 1/9*10)=1/2 * 22/45 = 11/45

2A = \(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+...+\frac{1}{8\cdot9}-\frac{1}{9\cdot10}\)

2A = \(\frac{1}{2}-\frac{1}{90}\)

2A = \(\frac{44}{90}\)

A = \(\frac{22}{90}\)