Ta có : 

\(M=\frac{5}{1.2.3}+\frac{5}{2.3.4}+...+\frac{5}{10.11.12}\)

\(M=5.\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{10.11.12}\right)\)

\(M=5.\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{10.11}-\frac{1}{11.12}\right)\)

\(M=\frac{5}{2}.\left(\frac{1}{1.2}-\frac{1}{11.12}\right)\)

\(M=\frac{5}{2}.\left(\frac{1}{2}-\frac{1}{132}\right)\)

\(M=\frac{5}{2}.\left(\frac{66}{132}-\frac{1}{132}\right)\)

\(M=\frac{5}{2}.\frac{65}{132}\)

\(M=\frac{325}{264}\)

Tham khảo nha !!! Chúc học tốt !!!

Công thức : 

\(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{6}\right)=\frac{1}{2}.\frac{1}{3}=\frac{1}{1.2.3}\)

đặt S=1.2.3+2.3.4+....+47.48.49

4S=1.2.3.(4-0)+2.3.4.(5-1)+...+47.48.49.(50-46)

4S=1.2.3.4-1.2.3+2.3.4.5-1.2.3.4+....+47.48.49.50-46.47.48.49

4S=47.48.49.50-1.2.3

S=(47.48.49.50-1.2.3):4

cool queen đúng rồi

???????????

\(S=1.2.3+2.3.4+...+98.99.100.\)

\(\Rightarrow4S=4\left(1.2.3+2.3.4+...+98.99.100\right)\)

\(=1.2.3.4+2.3.4.4+...+98.99.100.4\)

\(=1.2.3.\left(4-0\right)+2.3.4.\left(5-1\right)+...+98.99.100.\left(101-97\right)\)

\(=1.2.3.4-0.1.2.3+2.3.4.5-1.2.3.4+...+98.99.100.101-97.98.99.100\)

\(=98.99.100.101\)

\(\Rightarrow S=\frac{98.99.100.101}{4}=24497550\)

thanks

\(C=1.2.3+2.3.4+...+48.49.50\)

\(\Rightarrow4C=1.2.3.4+2.3.4.4+...+48.49.50.4\)

\(=1.2.3.4+2.3.4.\left(5-1\right)+...+48.49.50.\left(51-47\right)\)

\(=1.2.3.4+2.3.4.5-1.2.3.4+...+48.49.50.51-47.48.49.50\)

\(=48.49.50.51\)

\(\Rightarrow C=\frac{48.49.50.51}{4}=1499400\)

B= 1/ 1.2.3 + 1/ 2.3 4 + 1/ 3.4.5 + .... + 1/ 48.49.50

Mà ta có:

1/ 1.2 - 1/ 2.3 = 2/ 1.2.3

1/ 2.3 - 1/3.4 = 2/ 2.3.4

Từ đó=> B = 1/2 . ( 2/ 1.2.3 + 2/ 2,3.4 + ... + 2/ 18. 19. 20 )

= 1/2 .( 1/ 1.2 – 1/ 2.3 + 1/ 2.3 - .....- 1/19.20)

= 1/2. ( 1/ 1.2 – 1/ 19.20 ) = 1/ 2 . 189/380 = 189/760

\(\frac{18}{1.2.3}+\frac{18}{2.3.4}+...+\frac{18}{25.26.27}\)

\(=18.\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{25.26.27}\right)\)

\(=18.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{25.26}-\frac{1}{26.27}\right)\)

\(=18.\left(\frac{1}{1.2}-\frac{1}{26.27}\right)\)

\(=18.\left(\frac{1}{2}-\frac{1}{702}\right)\)

\(=18.\left(\frac{351}{702}-\frac{1}{702}\right)\)

\(=18.\frac{350}{702}\)

\(=\frac{350}{39}\)

=18x(1/1x2x3+1/2x3x4+1/3x4x5+...+1/25x26x27). =18x(1/1x2-1/2x3+1/2x3-1/3x4+...+1/25x26-1/26x27).                                                                 =18x(1/1x2-1/26x27).                                                =18x175/351.                                                             =350/39

Hai thằng sai cmnr

\(\frac{3}{1.2.3}+\frac{3}{2.3.4}+\frac{3}{3.4.5}+...+\frac{3}{15.16.17}\)

=\(\frac{3}{1.2.3}+\frac{3}{2.3.4}+\frac{3}{3.4.5}+...+\frac{3}{3.5.16.17}\)

=\(\frac{1}{1.2}+\frac{1}{2.4}+\frac{1}{4.5}+...+\frac{1}{5.16.17}\)

sai rồi bạn ơi!!! mk biết ko phải là như vậy!

\(A=\frac{3}{1.2.3}+\frac{3}{2.3.4}+\frac{3}{3.4.5}+...+\frac{3}{2015.2016.2017}\)

\(A=\frac{3}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\frac{3}{2}.\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+\frac{3}{2}.\left(\frac{1}{3.4}-\frac{1}{4.5}\right)+...+\frac{3}{2}.\left(\frac{1}{2015.2016}-\frac{1}{2016.2017}\right)\)

\(A=\frac{3}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2015.2016}-\frac{1}{2016.2017}\right)\)

\(A=\frac{3}{2}.\left(\frac{1}{1.2}-\frac{1}{2016.2017}\right)\)

\(A=\frac{3}{4}-\frac{3}{2.2016.2017}< 1\)