\(a.\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}\)

\(=\frac{1}{2}-\frac{1}{5}\)

\(=\frac{3}{10}\)

\(b.\frac{2}{2\cdot3}+\frac{2}{3\cdot4}+\frac{2}{4\cdot5}\)

\(=2\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}\right)\)

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

\(=2\cdot\left(\frac{1}{2}-\frac{1}{5}\right)\)

\(=2\cdot\frac{3}{10}=\frac{3}{5}\)

\(c.\frac{1}{2\cdot3}+\frac{2}{3\cdot5}+\frac{3}{5\cdot8}\)

\(=\frac{1}{6}+\frac{2}{15}+\frac{3}{40}\)

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

k nha 500 AE

a, \(\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}\)

\(=\frac{3-2}{2\times3}+\frac{4-3}{3\times4}+\frac{5-4}{4\times5}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}\)

\(=\frac{1}{2}-\frac{1}{5}\)

\(=\frac{3}{10}\)

b, \(\frac{2}{2\times3}+\frac{2}{3\times4}+\frac{2}{4\times5}\)

\(=\frac{3-2}{2\times3}+\frac{4-3}{3\times4}+\frac{5-4}{4\times5}\)

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

\(=\left(\frac{1}{2}-\frac{1}{5}\right)\times\frac{2}{1}\)

\(=\frac{3}{10}\times\frac{2}{1}\)

\(=\frac{3}{5}\)

c,  \(\frac{1}{2\times3}+\frac{2}{3\times5}+\frac{3}{5\times8}\)

 \(=\frac{3-2}{2\times3}+\frac{5-3}{3\times5}+\frac{8-5}{5\times8}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}\)

\(=\frac{1}{2}-\frac{1}{8}\)

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

Ta có: \(\frac{2}{2\times3}+\frac{2}{3\times4}+\cdots+\frac{2}{48\times49}+\frac{2}{49\times50}\)

\(=2\times\left(\frac{1}{2\times3}+\frac{1}{3\times4}+\cdots+\frac{1}{49\times50}\right)\)

\(=2\times\left(\frac12-\frac13+\frac13-\frac14+\cdots+\frac{1}{49}-\frac{1}{50}\right)\)

\(=2\times\left(\frac12-\frac{1}{50}\right)=1-\frac{1}{25}=\frac{24}{25}\)

dễ mà-TỰ LÀM nha pé

haiz khó thế

\(\frac{1}{2x3}+\frac{1}{3x4}+\frac{1}{4x5}+\frac{1}{5x4}\)

\(=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{20}=\frac{1}{60}+\frac{1}{60}+\frac{1}{60}+\frac{1}{60}=\frac{4}{60}=\frac{1}{15}\)

##

1/2.3+1/3.4+1/4.5+1/5.4

=1/2-1/3+1/3-1/4+1/4-1/5+1/20

=1/2-1/5+1/20

=3/10+1/20

=7/20

\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)

\(A=1-\frac{1}{100}=\frac{99}{100}\)

k cho mình nha bạn

=1-1/2+1/2-1/3+1/3-1/4+...+1/99-1/100

=1-1/100=99/100

Ta có : 

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}\)

\(=\)\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)

\(=\)\(1-\frac{1}{2014}\)

\(=\)\(\frac{2014}{2014}-\frac{1}{2014}\)

\(=\)\(\frac{2013}{2014}\)

Vậy \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}=\frac{2013}{2014}\)

Dấu \(.\) là dấu nhân nhé 

Chúc bạn học tốt ~

\(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{2013\times2014}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)

\(=1-\frac{1}{2014}\)

\(=\frac{2013}{2014}\)

CHÚC BN HỌC TỐT!!!!!
 

Mình không thể giải thích được nhưng kết quả chắc chắn là : \(\frac{8}{9}\)

đặt A=1/1×2+1/2×3+1/3×4+

Ta có: 1/1x2 + 1/2x3 + 1/3x4 +...+ 1/X x (X + 1) = 499/500

=> 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 +...+ 1/X - 1/(X + 1) = 499/500

=> 1 - 1/(X + 1) = 499/500

=>      1/(X + 1) = 1 - 499/500

=>      1/(X + 1) = 1/500

=>          X + 1 = 500

=>          X       = 500 - 1

=>          X       = 499 

Đáp số: X = 499

\(2\times\left(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+....+\frac{1}{18x19}+\frac{1}{19x20}\right)\)

\(2x\left(1-\frac{1}{20}\right)=2x\frac{19}{20}=\frac{19}{10}\)

\(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{18.19}+\frac{2}{19.20}\)

\(=1.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\right)\)

\(=1.\left(1-\frac{1}{20}\right)\)

\(=1.\frac{19}{20}\)

\(=\frac{19}{20}\)

Lưu ý: Từ bước thứ 2 bạn chuyển thành số La-tinh nhé.

P/s: "." là nhân nhé.