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[thieukhang61]

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[TEX]S = \frac{1}{{1.2.3}} + \frac{1}{{2.3.4}} + \frac{1}{{3.4.5}} + ... + \frac{1}{{8.9.10}}\[/TEX]

 

[huuthuyenrop2]

S= $\dfrac{1}{1.2.3} + \dfrac{1}{2.3.4} + ............+ \dfrac{1}{8.9.10}$
2S= $\dfrac{3-1}{1.2.3} + \dfrac{4-2}{2.3.4} + ............+ \dfrac{10-8}{8.9.10}$
2S= $\dfrac{1}{1.2} - \dfrac{1}{2.3}+ \dfrac{1}{2.3}-\dfrac{1}{3.4} + ............+ \dfrac{1}{8.9}-\dfrac{1}{9.10}$
2S= $\dfrac{1}{1.2} - \dfrac{1}{9.10}$
2S=$ \dfrac{22}{45}$
S=$\dfrac{11}{45}$

 

[0872]

$S = \dfrac{1}{{1.2.3}} + \dfrac{1}{{2.3.4}} + \dfrac{1}{{3.4.5}} + ... + \dfrac{1}{{8.9.10}}$

$S = \dfrac{1}{2}[\dfrac{1}{(1.2)} - \dfrac{1}{(2.3)}] + \dfrac{1}{2}[\dfrac{1}{(2.3)} - \dfrac{1}{(3.4)}] + \dfrac{1}{2}[\dfrac{1}{(3.4)} - \dfrac{1}{(4.5)}] + ... + \dfrac{1}{2}[\dfrac{1}{(8.9)} - \dfrac{1}{(9.10)}]$

$S = \dfrac{1}{2}[\dfrac{1}{(1.2)} - \dfrac{1}{(2.3)} + \dfrac{1}{(2.3)} - \dfrac{1}{(3.4)} + \dfrac{1}{(3.4)} - \dfrac{1}{(4.5)} + ... + \dfrac{1}{(8.9)} - \dfrac{1}{(9.10)}]$

$S = \dfrac{1}{2}[\dfrac{1}{(1.2)} - \dfrac{1}{(9.10)}]$

$S = \dfrac{1}{2}(\dfrac{1}{2} - \dfrac{1}{90})$

$S = \dfrac{1}{2}. \dfrac{22}{45}$

$S = \dfrac{11}{45}$

 

[23121999chien]

Hình như bài này làm thế này:
[TEX]S = \frac{1}{{1.2.3}} + \frac{1}{{2.3.4}} + \frac{1}{{3.4.5}} + ... + \frac{1}{{8.9.10}}\[/TEX]
Ta sẽ có: [TEX]2S = 2.(\frac{1}{{1.2.3}} +\frac{1}{{2.3.4}} + \frac{1}{{3.4.5}} + ... + \frac{1}{{8.9.10}})=\frac{2}{{1.2.3}} +\frac{2}{{2.3.4}} + \frac{2}{{3.4.5}} + ... + \frac{2}{{8.9.10}}\[/TEX]
2S=$\dfrac{1}{1.2}$-$\dfrac{1}{2.3}$+$\dfrac{1}{2.3}$-$\dfrac{1}{3.4}$+....+$\dfrac{1}{8.9}$-$\dfrac{1}{9.10}$
2S=$\dfrac{1}{1.2}$-$\dfrac{1}{9.10}$=$\dfrac{22}{45}$
=>S=$\dfrac{22}{45}$:2=$\dfrac{11}{45}$

 

[huy14112]

$S=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+\dfrac{1}{4.5.6}+...+\dfrac{1}{8.9.10}$

$2S=\dfrac{3-1}{1.2.3}+\dfrac{4-2}{2.3.4}+\dfrac{5-3}{3.4.5}+...+\dfrac{10-8}{8.9.10}$

$2S=\dfrac{3}{1.2.3}-\dfrac{1}{1.2.3}+\dfrac{4}{2.3.4}-\dfrac{2}{2.3.4}+\dfrac{5}{3.4.5}-\dfrac{3}{3.4.5}+...+\dfrac{10}{8.9.10}-\dfrac{8}{8.9.10}$

$2S=\dfrac{1}{1.2} - \dfrac{1}{2.3}+ \dfrac{1}{2.3}-\dfrac{1}{3.4} + ............+ \dfrac{1}{8.9}-\dfrac{1}{9.10}$

$2S=\dfrac{1}{1.2}-\dfrac{1}{9.10}$

$2S=\dfrac{1}{2}-\dfrac{1}{90}$

$2S=\dfrac{22}{45}$

$S=\dfrac{11}{45}$