hello 

Đặt \(A=\frac{4}{2.4}+\frac{4}{4.6}+\frac{4}{6.8}+...+\frac{4}{16.18}\)

\(A=\frac{4-2}{2.4}+\frac{6-4}{4.6}+\frac{8-6}{6.8}+....+\frac{18-16}{16.18}\)

\(A=\frac{4}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{16}-\frac{1}{18}\right)\)

\(A=\frac{4}{2}.\left(\frac{1}{2}-\frac{1}{18}\right)\)

\(A=\frac{4}{2}.\frac{4}{9}\)

\(\Rightarrow A=\frac{8}{9}\)

\(\frac{4}{2.4}+\frac{4}{4.6}+\frac{4}{6.8}+...+\frac{4}{16.18}\)

\(=\frac{4}{2}\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{16}-\frac{1}{18}\right)\)

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

\(=2.\frac{4}{9}\)

\(=\frac{8}{9}\)

I.\(B=9,8+8,7+7,6+...+2,1-1,2-2,3-3,4-...-8,9\)

\(B=\left(9,8-8,9\right)+\left(8,7-7,8\right)+\left(7,6-6,7\right)+...+\left(2,1-1,2\right)\)

\(B=0,9+0,9+0,9+...+0,9\) ( 8 số 0,9 )

\(B=7,2\)

II.

\(\left(a\right)\frac{2}{1\cdot2}+\frac{2}{2\cdot3}+\frac{2}{3\cdot4}+...+\frac{2}{19\cdot20}\)

\(=2\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{19\cdot20}\right)\)

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

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

\(=2\cdot\frac{19}{20}=\frac{19}{10}\)

\(\left(b\right)\frac{4}{1\cdot3}+\frac{4}{3\cdot5}+\frac{4}{5\cdot7}+...+\frac{4}{17\cdot19}+\frac{4}{19\cdot21}\)

\(=2\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{17\cdot19}+\frac{2}{19\cdot21}\right)\)

\(=2\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{17}-\frac{1}{19}+\frac{1}{19}-\frac{1}{21}\right)\)

\(=2\left(1-\frac{1}{21}\right)\)

\(=2\cdot\frac{20}{21}=\frac{40}{21}\)

\(\left(c\right)\frac{4}{2\cdot4}+\frac{4}{4\cdot6}+\frac{4}{6\cdot8}+...+\frac{4}{16\cdot18}+\frac{4}{18\cdot20}\)

\(=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{8\cdot9}+\frac{1}{9\cdot10}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}\)

\(=1-\frac{1}{10}=\frac{9}{10}\)

cảm ơn  bạn

1/ 2.4 + 1/4.6 + ...+1/18.20

= 1/2 - 1/4 + 1/4 -1/6 + .... + 1/18.20

trừ hết đi cho nhau cuối cùng:

= 1/2 - 1/20 = 9/20

$a)\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{90}$

$=\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{9.10}$

$=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{9}-\dfrac{1}{10}$

$=\dfrac{1}{2}-\dfrac{1}{10}=\dfrac{2}{5}$

b) Đặt $B=\dfrac{4}{2.4}+\dfrac{4}{4.6}+...+\dfrac{4}{18.20}$

$=>\dfrac{1}{2}B=\dfrac{2}{2.4}+\dfrac{2}{4.6}+...+\dfrac{2}{18.20}$

$=>\dfrac{1}{2}B=\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{18}-\dfrac{1}{20}$

$=>\dfrac{1}{2}B=\dfrac{1}{2}-\dfrac{1}{20}=\dfrac{9}{20}$

$=>B=\dfrac{9}{10}$

c) $\dfrac{2}{3}+\dfrac{2}{15}+\dfrac{2}{35}+\dfrac{2}{63}$

$=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}$

$=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}$

$=1-\dfrac{1}{9}=\dfrac{8}{9}$

d) Viết lại đề rõ ràng nha bạn.

Ta có: \(F=\dfrac{4}{2\cdot4}+\dfrac{4}{4\cdot6}+\dfrac{4}{6\cdot8}+...+\dfrac{4}{2008\cdot2010}\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2008}-\dfrac{1}{2010}\right)\)

\(=2\cdot\left(\dfrac{1}{2}-\dfrac{1}{2010}\right)\)

\(=2\cdot\dfrac{502}{1005}=\dfrac{1004}{1005}\)

\(F=\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2008.2010}\)

\(F=2.\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2008.2010}\right)\)

\(F=2.\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2008}-\dfrac{1}{2010}\right)\)

\(F=2.\left(\dfrac{1}{2}-\dfrac{1}{2010}\right)\)

\(F=1-\dfrac{1}{1005}=\dfrac{1004}{1005}\)

=2(2/2.4+2/4.6+......+22/008.2010)

=2(12−12010)

\(N=\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2014.2016}\)

\(=2\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2014.2016}\right)\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2014}-\dfrac{1}{2016}\right)\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{2016}\right)\)

\(=2\left(\dfrac{1008}{2016}-\dfrac{1}{2016}\right)\)

\(=2.\dfrac{1007}{2016}=\dfrac{1007}{1008}\)

Công thức đây bạn:

\(\dfrac{a}{n\left(n+a\right)}=\dfrac{1}{n}-\dfrac{1}{n+a}\)

Giải:

\(N=\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2014.2016}\) 

\(N=2.\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2014.2016}\right)\) 

\(N=2.\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2014}-\dfrac{1}{2016}\right)\) 

\(N=2.\left(\dfrac{1}{2}-\dfrac{1}{2016}\right)\) 

\(N=2.\dfrac{1007}{2016}\) 

\(N=\dfrac{1007}{1008}\) 

Công thức tính: \(\dfrac{1}{n.\left(n+1\right)}=\dfrac{1}{n}-\dfrac{1}{n+1}\) 

Chúc bạn học tốt!

K = 2( 2/2.4 + 2/4.6 +......+ 2/2008.2010)

K = 2( 1/2 - 1/4 + 1/4 - 1/6 +......+ 1/2008 - 1/2010)

K = 2( 1/2 - 1/2010)

K = 2 . 1004/2010

K = 1004/1005

Ai k mk mk k lại 

K=\(\frac{4}{2.4}+\frac{4}{4.6}+\frac{4}{6.8}+...+\frac{4}{2008.2010}\)

K=\(\frac{4}{2}.\frac{2}{2.4}+\frac{4}{2}.\frac{2}{4.6}+...+\frac{4}{2}.\frac{2}{2008.2010}\)

K=\(\frac{4}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{2008.2010}\right)\)

K=\(\frac{4}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+..+\frac{1}{2008}-\frac{1}{2010}\right)\)

K=\(\frac{4}{2}.\left(\frac{1}{2}-\frac{1}{2010}\right)\)

K=\(\frac{4}{2}.\frac{502}{1005}\)

K=\(\frac{1004}{1005}\)