\(\Rightarrow C=\frac{10}{5}\left(\frac{1}{7.12}+\frac{1}{12.17}+\frac{1}{17.22}+...+\frac{1}{502.507}\right)\)

\(\Rightarrow C=2.\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+....+\frac{1}{507}-\frac{1}{507}\right)\)

\(\Rightarrow C=2.\left(\frac{1}{7}-\frac{1}{507}\right)=2.\frac{1}{7}-2.\frac{1}{507}=\frac{2}{7}-\frac{2}{507}\)

Cho S=\(\frac{5}{2.7}+\frac{5}{7.12}+...+\frac{5}{22.27}\)

\(=\frac{1}{2}-\frac{1}{7}+\frac{1}{7}-\frac{1}{12}+...+\frac{1}{22}-\frac{1}{27}\)

\(=\frac{1}{2}-\frac{1}{27}=\frac{25}{54}\)

kb nha mn!

\(\frac{5}{2}-\frac{5}{7}+\frac{5}{7}-\frac{5}{12}+\frac{5}{12}-\frac{5}{17}+\frac{5}{17}-\frac{5}{22}+\frac{5}{22}-\frac{5}{29}=\frac{5}{2}-0-0-0-0-\frac{5}{29}=\frac{5}{2}-\frac{5}{29}=\frac{145}{58}-\frac{10}{58}=\frac{135}{58}\)

Đặt
\(A=\frac{5}{2.7}+\frac{5}{7.12}+\frac{5}{12.17}+\frac{5}{17.22}+\frac{5}{22.27}+\frac{5}{27.32}+\frac{5}{32.37}\)

\(A=\frac{1}{2}-\frac{1}{7}+\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+\frac{1}{17}-\frac{1}{22}+\frac{1}{22}-\frac{1}{27}+\frac{1}{27}-\frac{1}{32}+\frac{1}{32}-\frac{1}{37}\)

\(A=\frac{1}{2}-\frac{1}{37}=\frac{35}{74}\)

\(=\frac{1}{2}-\frac{1}{7}+\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+...+\frac{1}{32}-\frac{1}{37}\)

\(=\frac{1}{2}-\frac{1}{37}\)

\(=\frac{37}{74}-\frac{2}{74}=\frac{35}{74}\)

\(=\frac{1}{2}.\frac{1}{7}+\frac{1}{7}.\frac{1}{12}...+\frac{1}{32}.\frac{1}{37}\)

= \(\frac{1}{2}+\frac{1}{37}\)

= \(\frac{37}{74}+\frac{2}{74}\)

\(\frac{39}{74}\)

tíck mik nha, mik nhanh nhất

\(B=2\left(\frac{5}{7.12}+\frac{5}{12.17}+...+\frac{5}{502.507}\right)\)

\(B=2\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+...+\frac{1}{502}-\frac{1}{507}\right)\)

\(B=2\left(\frac{1}{7}-\frac{1}{507}\right)\)

\(B=2\times\frac{500}{3549}\)

\(B=\frac{1000}{3549}\)

\(B=\frac{10}{7.12}+\frac{10}{12.17}+\frac{10}{17.22}+......+\frac{10}{502.507}\)

\(B=\frac{10}{5}.\left(\frac{5}{7.12}+\frac{5}{12.17}+\frac{5}{17.22}+........+\frac{5}{502.507}\right)\)

\(B=\frac{10}{5}.\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+\frac{1}{17}-\frac{1}{22}+.....+\frac{1}{502}-\frac{1}{507}\right)\)

\(B=\frac{10}{5}.\left(\frac{1}{7}-\frac{1}{507}\right)=2.\frac{500}{3549}=\frac{1000}{3549}\)

\(K=\frac{10}{7.12}+\frac{10}{12.17}+\frac{10}{17.22}+...+\frac{10}{502.507}\)

\(\Leftrightarrow K=\frac{10}{5}\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+\frac{1}{17}-\frac{1}{22}+...+\frac{1}{502}-\frac{1}{507}\right)\)

\(\Leftrightarrow K=2\left(\frac{1}{7}-\frac{1}{507}\right)\)

\(\Leftrightarrow K=2\cdot\frac{500}{3549}\)

\(\Leftrightarrow K=\frac{1000}{3549}\)

5xK=5/7.12+5/12.17+..............+5/502.507

5xK=(1/7-1/12)+(1/12-1/17)+........+(1/502-1/507)

5xK=1/7-1/507

5xK=500/3549

K   =500/3549:5

K    =100/3549

Đặt A=đã cho

=>\(\frac{5}{3}A=\frac{5}{2\cdot7}+\frac{5}{7\cdot12}+...+\frac{5}{2012\cdot2017}\)

=>\(\frac{5}{3}A=\frac{1}{2}-\frac{1}{2017}\)

Đến đây dễ rồi tự lm tiếp nhé
 

\(=\frac{3}{5}.\left(\frac{5}{2.7}+\frac{5}{7.12}+\frac{5}{12.17}+...+\frac{5}{2012.2017}\right)\)

\(=\frac{3}{5}.\left(\frac{1}{2}-\frac{1}{7}+\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+...+\frac{1}{2012}-\frac{1}{2017}\right)\)

\(=\frac{3}{5}.\left(\frac{1}{2}-\frac{1}{2017}\right)\)

\(=\frac{3}{5}.\left(\frac{2015}{4034}\right)\)

\(=\frac{1209}{4034}\)

Bài này bạn phải học lý thuyết mới làm được nhé!! Chúc bạn zui~^^

ko hiu lun =.=

\(B=\frac{10}{7.12}+\frac{10}{12.17}+\frac{10}{17.22}+...+\frac{10}{2017.2022}\)

\(=2\left(\frac{5}{7.12}+\frac{5}{12.17}+\frac{5}{17.22}+...+\frac{5}{2017.2022}\right)\)

\(=2\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+...+\frac{1}{2017}-\frac{1}{2022}\right)\)

\(=2\left(\frac{1}{7}-\frac{1}{2022}\right)=\frac{2015}{7077}\)

\(B=\frac{10}{7.12}+\frac{10}{12.17}+\frac{10}{17.22}+...+\frac{10}{2017.2022}\)

\(\Rightarrow B=2.\left(\frac{5}{7.12}+\frac{5}{12.17}+\frac{5}{17.22}+...+\frac{5}{2017.2022}\right)\)

\(\Rightarrow B=2.\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+\frac{1}{17}-\frac{1}{22}+...+\frac{1}{2017}-\frac{1}{2022}\right)\)

\(\Rightarrow B=2.\left(\frac{1}{7}-\frac{1}{2022}\right)\)

\(\Rightarrow B=2.\frac{2015}{14154}\)

\(\Rightarrow B=\frac{2015}{7077}.\)

Vậy \(B=\frac{2015}{7077}.\)

Chúc bạn học tốt!

\(A=\frac{1}{7\cdot12}+\frac{1}{12\cdot17}+\frac{1}{17\cdot22}+...+\frac{1}{52\cdot57}\)

\(A=\frac{1}{5}\left(\frac{5}{7\cdot12}+\frac{5}{12\cdot17}+\frac{5}{17\cdot22}+...+\frac{5}{52\cdot57}\right)\)

\(A=\frac{1}{5}\left(\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{17}+...+\frac{1}{52}-\frac{1}{57}\right)\)

\(A=\frac{1}{5}\left(\frac{1}{7}-\frac{1}{57}\right)=\frac{1}{5}\cdot\frac{50}{399}=\frac{10}{399}\)

\(B=\frac{10}{8\cdot13}+\frac{10}{13\cdot18}+\frac{10}{18\cdot23}+...+\frac{10}{253\cdot258}\)

\(B=\frac{10}{5}\left(\frac{5}{8\cdot13}+\frac{5}{13\cdot18}+\frac{5}{18\cdot23}+...+\frac{5}{253\cdot258}\right)\)

\(B=2\left(\frac{1}{8}-\frac{1}{13}+\frac{1}{13}-\frac{1}{18}+...+\frac{1}{253}-\frac{1}{258}\right)\)

\(B=2\left(\frac{1}{8}-\frac{1}{258}\right)=2\cdot\frac{125}{1032}=\frac{125}{516}\)

*Cái đây giải thích hơi bị " khó hiểu " :

Chỗ mẫu (12 - 7) = (17 - 12) = ... = (57 - 52) = 5

Tử là 1 , mẫu là 5 nên tử/mẫu = 1/5

Hay \(\frac{1}{5}\left(\frac{5}{7\cdot12}+\frac{5}{12\cdot17}+...+\frac{5}{52\cdot57}\right)\)

Còn bạn Trương Bùi Linh thì :

Mẫu = (13 - 8) = (18 - 13) = (23 - 18) = ... = 5

Tử là 10,mẫu là 5 => tử / mẫu = 10/5 = 2