\(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+...+\frac{4}{177.181}+\frac{4}{181.185}\)

\(=\left(\frac{1}{1}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{9}\right)+\left(\frac{1}{9}-\frac{1}{13}\right)+...+\left(\frac{1}{177}-\frac{1}{181}\right)+\left(\frac{1}{181}-\frac{1}{185}\right)\)

\(=\frac{1}{1}-\frac{1}{185}\)

\(=\frac{184}{185}\)

a)   \(\frac{1}{2\cdot4}+\frac{1}{4\cdot6}+\frac{1}{6\cdot8}+\frac{1}{8\cdot10}\)

\(=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+\frac{1}{8}-\frac{1}{10}\right)\)

\(=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{10}\right)\)

\(=\frac{1}{2}\cdot\frac{2}{5}=\frac{2}{10}=\frac{1}{5}\)

b)   \(\frac{4}{1\cdot5}+\frac{4}{5\cdot9}+\frac{4}{9\cdot13}+\frac{4}{13\cdot17}\)

\(=1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\frac{1}{13}-\frac{1}{17}\)

\(=1-\frac{1}{17}=\frac{16}{17}\)

hok tốt ...

a)\(A=\frac{1}{2\cdot4}+\frac{1}{4\cdot6}+\frac{1}{6\cdot8}+\frac{1}{8\cdot10}\)

\(2A=\frac{2}{2\cdot4}+\frac{2}{4\cdot6}+\frac{2}{6\cdot8}+\frac{2}{8\cdot10}\)

\(2A=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+\frac{1}{8}-\frac{1}{10}=\frac{1}{2}-\frac{1}{10}=\frac{2}{5}\)

\(A=\frac{2}{5}\cdot\frac{1}{2}=\frac{1}{5}\)

b)\(B=\frac{4}{1\cdot5}+\frac{4}{5\cdot9}+\frac{4}{9\cdot13}+\frac{4}{13\cdot17}=1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\frac{1}{13}-\frac{1}{17}=1-\frac{1}{17}=\frac{16}{17}\)

a) 1/2.4 + 1/4.6 + 1/6.8 + 1/8.10

= 2/2.2.4 + 2/2.4.6 + 2/2.6.8 + 2/2.8.10 ( nhân cả tử và mẫu với 2)

= 1/2 .( 2/2.4 + 2/4.6 + 2/6.8 + 2/8.10 )

= 1/2 .(1/2 - 1/4 + 1/4 - 1/8 + 1/8 - 1/10)

= 1/2.(1/2 - 1/10)

= 1/2.( 5/10 - 1/10) = 1/2.4/10 = 2/10 = 1/5

b) 4/1.5+ 4/5.9 + 4/ 9.13 + 4/13.17

= 1- 1/5 + 1/5 - 1/9 + 1/9 - 1/13 + 1/13 - 1/17

= 1- 1/17

= 16/17 

4/1.5+4/5.9+4/9.13+....+4/21.25

=1-1/5+1/5-1/9+1/9-1/13+......+1/21-1/25

=1-1/25

=24/25

Tích đúng cho mình nha

\(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+....+\frac{4}{21.25}\)

\(=1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+....+\frac{1}{21}-\frac{1}{25}\)

\(=1-\frac{1}{25}=\frac{24}{25}\)

\(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+...+\frac{4}{21.25}\)

=\(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+...+\frac{1}{21}-\frac{1}{25}\)

=\(1-\frac{1}{25}\)

=\(\frac{24}{25}\)

Chúc bạn học giỏi nha!!!

Có dạng tổng quát như thế này nhé: 
\(\frac{k}{n\left(n+k\right)}=\frac{1}{n}-\frac{1}{k+n}\)

Trong trường hợp này là \(\frac{-4}{1.5}-\frac{4}{5.9}-...-\frac{4}{\left(n+4\right)n}=-\left(\frac{1}{1}-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+...+\frac{1}{n}-\frac{1}{n+4}\right)\)

Đáp án là: \(\frac{1}{n+4}-1\)

\(D=4\left(\frac{4}{1.5}+\frac{4}{5.9}+...+\frac{4}{201.205}\right)\)
\(D=4\left(\left(1-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{9}\right)+...+\left(\frac{1}{201}-\frac{1}{205}\right)\right)\)
D=4[(1-1/205)
D=4.204/205
=>D=816/205
____________________--
li-ke cho mình nhé bn Cao Minh Hoàng

\(A=\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{17\cdot21}< 1\)

\(A=\dfrac{4}{4}\cdot\left(\dfrac{1}{1\cdot5}+\dfrac{1}{5\cdot9}+\dfrac{1}{9\cdot13}+...+\dfrac{1}{17\cdot21}\right)< 1\)

\(A=\dfrac{1}{1}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{17}-\dfrac{1}{21}< 1\)

\(A=1-\dfrac{1}{21}< 1\) (đúng) (đpcm).

Đề sai

Ta có: \(\dfrac{4}{1.5}+\dfrac{4}{5.9}+\dfrac{4}{9.13}+...+\dfrac{4}{17.21}\)

=\(\dfrac{4}{4}.\left(\dfrac{1}{1.5}+\dfrac{1}{5.9}+\dfrac{1}{9.13}+...+\dfrac{1}{17.21}\right)\)

=\(1\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{17}-\dfrac{1}{21}\right)\)

=\(1-\dfrac{1}{21}\)

mà \(1-\dfrac{1}{21}\)<1

=>\(\dfrac{4}{1.5}+\dfrac{4}{5.9}+\dfrac{4}{9.13}+...+\dfrac{4}{17.21}\)<1