Đang cần gấp ạ.

Ta có :

 1/2x6 + 1/4x9 + 1/6x12 +...+1/198 x 300

 = 1/6x2 + 1/6x6 + 1/6x12 + ....+1/6x9900

 = 1/6 x ( 1/2 + 1/6 + 1/ 12 +...+1/9900)

 = 1/6 x (1/1x2 + 1/2x3 + 1/3x4+...+1/99x100)

 =1/6x (1-1/2 + 1/2-1/3 + 1/3 - 1/4 + ....+1/99-1/100)

 =1/6x(1-1/100)

 =1/6 x 99/100

 = 33/200

k cho mình nha , học tốt

A = \(\dfrac{1}{3\times6}\) + \(\dfrac{1}{6\times9}\) + \(\dfrac{1}{9\times12}\)+...+\(\dfrac{1}{144\times147}\)

A = \(\dfrac{1}{3}\) \(\times\)( \(\dfrac{3}{3\times6}\) + \(\dfrac{3}{6\times9}\)+\(\dfrac{1}{9\times12}\)+...+\(\dfrac{3}{144\times147}\))

A = \(\dfrac{1}{3}\) \(\times\)(\(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{12}+...+\dfrac{1}{144}-\dfrac{1}{147}\))

A = \(\dfrac{1}{3}\)\(\times\)(\(\dfrac{1}{3}\) - \(\dfrac{1}{147}\))

A = \(\dfrac{1}{3}\) \(\times\)\(\dfrac{16}{49}\)

A = \(\dfrac{16}{147}\)

Ta có: \(\frac{1}{4\times6}=\frac{1}{4\times1\times3\times2}=\frac{1}{4\times3\times1\times2}\)

\(\frac{1}{8\times9}=\frac{1}{4\times2\times3\times3}=\frac{1}{4\times3\times2\times3}\)

\(\frac{1}{12\times12}=\frac{1}{4\times3\times3\times4}\)

\(\frac{1}{16\times15}=\frac{1}{4\times4\times3\times5}=\frac{1}{4\times3\times4\times5}\)......

\(\frac{1}{2680\times2013}=\frac{1}{4\times670\times3\times671}\)

Do đó:

\(M=\frac{1}{4\times3}\times\left(\frac{1}{1\times2}+\frac{1}{2\times3}+....+\frac{1}{670\times671}\right)\)

\(=\frac{1}{12}\times\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{670}-\frac{1}{671}\right)\)

\(=\frac{1}{12}\times\left(\frac{1}{1}-\frac{1}{671}\right)=\frac{1}{12}\times\frac{670}{671}=\frac{335}{4026}\)

Vậy \(M=\frac{335}{4026}\)

S = 1/5.6 + 1/10.9+....+ 1/3350.2013

=1/5 . 1/3 .( 1/2+ 1/2.3 + 1/3.4 +... + 1/670.671)

=1/15. ( 1-1/2 + 1/2 - 1/3+...+ 1/670-1/671)

= 1/15 .( 1 - 1/671 )

= 1/15 .670/671

=134/2013

S = \(\frac{1}{5.6}+\frac{1}{10.9}+\frac{1}{15.12}+...+\frac{1}{3350.2013}=\frac{1}{5.3}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{670.671}\right)\)

\(=\frac{1}{15}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{670}-\frac{1}{671}\right)=\frac{1}{15}\left(1-\frac{1}{671}\right)=\frac{1}{15}.\frac{670}{671}\)

\(=\frac{134}{2013}\)

S = 1/5.6 + 1/10.9+....+ 1/3350.2013

=1/5 . 1/3 .( 1/2+ 1/2.3 + 1/3.4 +... + 1/670.671)

=1/15. ( 1-1/2 + 1/2 - 1/3+...+ 1/670-1/671)

= 1/15 .( 1 - 1/671 )

= 1/15 .670/671

=134/2013

S = 1/5.6 + 1/10.9+....+ 1/3350.2013

=1/5 . 1/3 .( 1/2+ 1/2.3 + 1/3.4 +... + 1/670.671)

=1/15. ( 1-1/2 + 1/2 - 1/3+...+ 1/670-1/671)

= 1/15 .( 1 - 1/671 )

= 1/15 .670/671

=134/2013

S = 1/5.6 + 1/10.9+....+ 1/3350.2013

=1/5 . 1/3 .( 1/2+ 1/2.3 + 1/3.4 +... + 1/670.671)

=1/15. ( 1-1/2 + 1/2 - 1/3+...+ 1/670-1/671)

= 1/15 .( 1 - 1/671 )

= 1/15 .670/671

=134/2013

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{8.9.10}=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\frac{1}{2}.\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\frac{1}{2}.\left(\frac{1}{8.9}-\frac{1}{9.10}\right)\)

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

\(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{9.10}\right)=\frac{1}{2}.\frac{22}{45}=\frac{11}{45}\)

\(\frac{4}{3.6}+\frac{4}{6.9}+\frac{4}{9.12}+\frac{4}{12.15}\)

\(=\frac{4}{3}.\left(\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+\frac{3}{12.15}\right)\)

\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+\frac{1}{12}-\frac{1}{15}\right)\)

\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{15}\right)\)

\(=\frac{4}{3}.\left(\frac{5}{15}-\frac{1}{15}\right)\)

\(=\frac{4}{3}.\frac{4}{15}=\frac{16}{45}\)

Dấu . là nhân nha

\(\frac{4}{3.6}+\frac{4}{6.9}+\frac{4}{9.12}+\frac{4}{12.15}\)

\(=\frac{4}{3}.\left(\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+\frac{3}{12.15}\right)\)

\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+\frac{1}{12}-\frac{1}{15}\right)\)

\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{15}\right)\)

\(=\frac{4}{3}.\frac{4}{15}=\frac{16}{45}\)

\(A=\frac{4}{3.6}+\frac{4}{6.9}+\frac{4}{9.12}+\frac{4}{12.15}\)

\(=>\frac{3}{4}A=\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+\frac{3}{12.15}\)

\(=>\frac{3}{4}A=\left(\frac{1}{3}-\frac{1}{6}\right)+\left(\frac{1}{6}-\frac{1}{9}\right)+\left(\frac{1}{9}-\frac{1}{12}\right)+\left(\frac{1}{12}-\frac{1}{15}\right)\)

\(=>\frac{3}{4}A=\frac{1}{3}-\frac{1}{15}\)

\(=>\frac{3}{4}A=\frac{4}{15}\)

\(=>A=\frac{4}{15}:\frac{3}{4}\)

\(=>A=\frac{4}{45}\)