\(\frac{1}{10}+\frac{1}{30}+\frac{1}{60}+\frac{1}{100}+\frac{1}{150}\)

= \(\frac{1}{10}.\left(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}\right)\)

= \(\frac{1}{10}.2.\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}\right)\)

= \(\frac{1}{5}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}\right)\)

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

= \(\frac{1}{5}.\left(1-\frac{1}{6}\right)\)

= \(\frac{1}{5}.\frac{5}{6}\)

= \(\frac{1}{6}\)

\(\frac{1}{10}+\frac{1}{30}+\frac{1}{60}+\frac{1}{100}+\frac{1}{150}\)

\(=\frac{1}{10}.\left(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}\right)\)

\(=\frac{1}{10}.2.\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}\right)\)

\(=\frac{1}{10}.2.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}\right)\)

\(=\frac{1}{10}.2.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\right)\)

\(=\frac{1}{5}.\left(1-\frac{1}{6}\right)\)

\(=\frac{1}{5}.\frac{5}{6}\)

\(=\frac{1}{6}\)

Rất vui vì giúp đc bạn <3

đặt 1/6 chung rồi giải 

a: \(D=\frac{10}{100}+\frac{10}{150}+\frac{10}{210}+\cdots+\frac{10}{1200}\)

\(=\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+\cdots+\frac{1}{120}\)

\(=\frac{2}{20}+\frac{2}{30}+\cdots+\frac{2}{240}=2\left(\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\cdots+\frac{1}{15\cdot16}\right)\)

\(=2\left(\frac14-\frac15+\frac15-\frac16+\cdots+\frac{1}{15}-\frac{1}{16}\right)=2\left(\frac14-\frac{1}{16}\right)=2\cdot\frac{3}{16}=\frac38\)

b: \(E=1\cdot2+2\cdot3+\cdots+99\cdot100\)

\(=1\left(1+1\right)+2\left(2+1\right)+\cdots+99\left(99+1\right)\)

\(=\left(1^2+2^2+\cdots+99^2\right)+\left(1+2+\cdots+99\right)\)

\(=\frac{99\left(99+1\right)\left(2\cdot99+1\right)}{6}+\frac{99\left(99+1\right)}{2}=\frac{99\cdot100\cdot199}{6}+99\cdot\frac{100}{2}\)

\(=33\cdot50\cdot199+99\cdot50\)

\(=33\cdot50\cdot\left(199+3\right)=33\cdot50\cdot202=33\cdot101\cdot100=333300\)

c: \(F=1^2+2^2+\cdots+98^2\)

\(=\frac{98\left(98+1\right)\left(2\cdot98+1\right)}{6}=\frac{98\cdot99\cdot197}{6}=49\cdot33\cdot197=318549\)

A = \(\frac{2}{3}\). \(\frac{5}{6}\). \(\frac{9}{10}\).\(\frac{14}{15}\)... \(\frac{209}{210}\)

  = \(\frac{4}{6}\). \(\frac{10}{12}\).\(\frac{18}{20}\).\(\frac{28}{30}\)... \(\frac{418}{420}\)

  = \(\frac{4.10.18.28...418}{6.12.20.30...420}\)

  =  \(\frac{1.4.2.5.3.6.4.7...19.22}{2.3.3.4.4.5.5.6...20.21}\)

  = \(\frac{\left(1.2.3.4...19\right)\left(4.5.6.7...22\right)}{\left(2.3.4.5...20\right)\left(3.4.5.6...21\right)}\)

  =  \(\frac{1.22}{20.3}\)= \(\frac{22}{60}\)= \(\frac{11}{30}\)

=14\15.20\21....209\210

=14.2\15.2.20.2\21.2....209.2\210.2

=4.7\5.6.5.8\6.7.....19.20\20.21

=4.5.6....19\5.6.7...20.7.8.9....22\6.7.8.....21

=4\20.22\6

=1\5-11\3

=11\15

k cho mik nha

Có chắc ko đó bạn

\(S=1+\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+...+\frac{101}{2}\)

\(S=1+\frac{1+2+3+4+...+101}{2}\)

\(S=1+\frac{10201}{2}=...\)

tick cho mink nha!

Ta có:\(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{1}{210}\) 

   \(=\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+....+\frac{1}{14.15}\)

    \(=\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+...+\frac{1}{14}-\frac{1}{15}\)

      \(=\frac{1}{6}-\frac{1}{15}=\frac{1}{10}\)

gọi biểu thức tên là A , ta có :

A= (1/21+1/210+1/2010). ( 1/3-1/30-1/5-1/10)

A = (1/21+1/210+1/2010) . 0

A = 0

  Công Chúa Giá Băng    làm đúng nhất !