1/10+1/15+1/21+...+1/120

=2*(1/20+1/30+1/42+...+1/240)

=2*(1/4*5+1/5*6+...+1/15*16)

=2*(1/4-1/5+1/5-1/6+...+1/15-1/16)

=2*[(1/4-1/16)+(1/5-1.5)+...+(1/15-1/15)]

=2[(4/16-1/16)+0+...+0]]

=2*3/16=3/8

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

\(S=\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+....+\frac{2}{240}\)

\(2S=\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+....+\frac{1}{240}\)

\(2S=\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+.....+\frac{1}{15.16}\)

\(2S=\left(\frac{1}{4}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)+\left(\frac{1}{6}-\frac{1}{7}\right)+.....+\left(\frac{1}{15}-\frac{1}{16}\right)\)

\(2S=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+....+\frac{1}{15}-\frac{1}{16}\)

\(2S=\frac{1}{4}-\frac{1}{16}\)

\(2S=\frac{3}{16}\)

\(S=\frac{3}{8}\)

= 1 : 10 + 1 : 15 + 1 : 21 + ... + 1 : 120

= 1 : (10 + 15 + 21 + ... + 120)

= 1 : 670 = 1/670

 Đặt A = \(\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+....+\frac{1}{120}\) 

=> A = \(2\left(\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+....+\frac{1}{240}\right)\)

        = \(2\left(\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+....+\frac{1}{15.16}\right)\)

        = \(2\left(\frac{1}{4}-\frac{1}{16}\right)=2\left(\frac{4}{16}-\frac{1}{16}\right)=2.\frac{3}{16}=\frac{3}{8}\)

Đặt A=1/10+1/15+1/21+...+1/120

    1/2 A=1/20+1/30+1/42+...+1/240

     A=1/4-1/5+1/5-1/6+1/6-1/7+...+1/15-1/16

     A=1/4-1/16

     A=3/16

Vậy:1/10+1/15+1/21+...+1/120=3/16

\(C=\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+...+\frac{2}{240}=2\times\left(\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{240}\right)\)

\(C=2\times\left(\frac{1}{4\times5}+\frac{1}{5\times6}+\frac{1}{6\times7}+...+\frac{1}{15\times16}\right)\)

\(C=2\times\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{15}-\frac{1}{16}\right)=2\times\left(\frac{1}{4}-\frac{1}{16}\right)=\frac{3}{8}\)

\(\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{120}=\frac{1}{2}x\left(\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{120}\right):\frac{1}{2}\)

 = \(\left(\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{240}\right):\frac{1}{2}=\left(\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{15.16}\right):\frac{1}{2}\)

= \(\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{15}-\frac{1}{16}\right):\frac{1}{2}=\left(\frac{1}{4}-\frac{1}{16}\right):\frac{1}{2}=\frac{3}{8}\)

\(\frac{1}{2}B=\frac{1}{20}+\frac{1}{30}+...+\frac{1}{240}\)

\(\frac{1}{2}B=\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{15.16}\)

\(\frac{1}{2}B=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{15}-\frac{1}{16}\)

\(\frac{1}{2}B=\frac{1}{4}-\frac{1}{16}=\frac{3}{16}\Rightarrow B=\frac{3}{16}:\frac{1}{2}=\frac{3}{8}\)

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

\(B=\frac{2}{4.5}+\frac{2}{5.6}+\frac{2}{6.7}+...+\frac{2}{15.16}\)

\(B=2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{15}-\frac{1}{16}\right)\)

\(B=2\left(\frac{1}{4}-\frac{1}{16}\right)\)

\(B=2.\frac{3}{16}=\frac{6}{16}=\frac{3}{8}\)

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

\(\Rightarrow\frac{1}{2}B=\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{240}\)

\(\Rightarrow\frac{1}{2}B=\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{15.16}\)

\(\Rightarrow\frac{1}{2}B=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{15}-\frac{1}{16}\)

\(\Rightarrow\frac{1}{2}B=\frac{1}{4}-\frac{1}{16}\)

\(\Rightarrow\frac{1}{2}B=\frac{3}{16}\)

\(\Rightarrow B=\frac{3}{16}\div\frac{1}{2}\)

\(\Rightarrow B=\frac{3}{8}\)

=2/20+2/10+2/42+..+2/240=2.(1/20+1/30+1/42+...+1/240)

=2.(1/4.5+1/5.6+1/6.7+...+1/15.16)

=2.(1/4-1/5+1/5-1/6+1/6-1/7+...+1/15-1/16)=2.(1/4-1/16)

=3/8 nhé

CHÚC BẠN HỌC GIỎI

K MÌNH NHÉ

À mình quên chỗ 1/5 phải là 1/15 nha bạn

Ta có:

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

\(C.\frac{1}{2}=\left(\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{120}\right).\frac{1}{2}\)

\(C.\frac{1}{2}=\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{240}\)

\(\frac{1}{2}C=\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+..+\frac{1}{15.16}\)

\(\frac{1}{2}C=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{15}-\frac{1}{16}\)

\(\frac{1}{2}C=\frac{1}{4}-\frac{1}{16}=\frac{3}{16}\)

\(\Rightarrow C=\frac{3}{8}\)

y = 1/10 + 1/15 + 1/21 + ... + 1/120

1/2y = 1/20 + 1/30 + 1/42 + ... + 1/240

1/2y = 1/4x5 + 1/5x6 + 1/6x7 + ... + 1/15x16

1/2y = 1/4 -1/5 + 1/5 - 1/6 + 1/6 -1/7 + ... +1/15 -1/16

1/2y = 1/4 - 1/16

1/2y = 3/16

y = 3/16 : 1/2

y = 3/16 x 2

y= 3/8 

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

\(y=\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+...+\frac{2}{240}\)

\(y=\frac{2}{4.5}+\frac{2}{5.6}+\frac{2}{6.7}+...+\frac{2}{15.16}\)

\(y=2.\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{15}-\frac{1}{16}\right)\)

\(y=2.\left(\frac{1}{4}-\frac{1}{16}\right)\)

\(y=2.\frac{3}{16}\)

\(y=\frac{3}{8}\)

\(\Rightarrow y=2.\left(\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+.........+\frac{1}{240}\right)\)

\(\Rightarrow y=2.\left(\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+.............+\frac{1}{15.16}\right)\)

\(\Rightarrow y=2.\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...........+\frac{1}{15}-\frac{1}{16}\right)\)

\(\Rightarrow y=2.\left(\frac{1}{4}-\frac{1}{16}\right)\)

\(\Rightarrow y=2.\frac{3}{16}\)

\(\Rightarrow y=\frac{3}{8}\)