=(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9) chia 2 

=(1-1/9)chia 2

=8/9 chia 2

=4/9

Đặt $A=\genfrac{}{}{0ex}{}{1}{1x3}+\genfrac{}{}{0ex}{}{1}{3x5}+\genfrac{}{}{0ex}{}{1}{5x7}+\genfrac{}{}{0ex}{}{1}{7x9}$

$2A=\genfrac{}{}{0ex}{}{2}{1x3}+\genfrac{}{}{0ex}{}{2}{3x5}+\genfrac{}{}{0ex}{}{2}{5x7}+\genfrac{}{}{0ex}{}{2}{7x9}$

$2A=\genfrac{}{}{0ex}{}{1}{1}-\genfrac{}{}{0ex}{}{1}{3}+\genfrac{}{}{0ex}{}{1}{3}-\genfrac{}{}{0ex}{}{1}{5}+...+\genfrac{}{}{0ex}{}{1}{7}-\genfrac{}{}{0ex}{}{1}{9}$

$2A=\genfrac{}{}{0ex}{}{1}{1}-\genfrac{}{}{0ex}{}{1}{9}=\genfrac{}{}{0ex}{}{8}{9}$

$A=\genfrac{}{}{0ex}{}{8}{9}.\genfrac{}{}{0ex}{}{1}{2}=\genfrac{}{}{0ex}{}{4}{9}$
 

A = \(\dfrac{1}{3\times5}\) + \(\dfrac{1}{5\times7}\) + \(\dfrac{1}{7\times9}\)+...+ \(\dfrac{1}{2009\times2011}\)

A = \(\dfrac{1}{2}\) \(\times\) ( \(\dfrac{2}{3\times5}\) + \(\dfrac{2}{5\times7}\)+ \(\dfrac{2}{7\times9}\)+...+ \(\dfrac{1}{2009\times2011}\))

A = \(\dfrac{1}{2}\) \(\times\) ( \(\dfrac{1}{3}\) - \(\dfrac{1}{5}\) + \(\dfrac{1}{5}\) - \(\dfrac{1}{7}\) + \(\dfrac{1}{7}\) - \(\dfrac{1}{9}\)+...+ \(\dfrac{1}{2009}\) - \(\dfrac{1}{2011}\))

A = \(\dfrac{1}{2}\) \(\times\) ( \(\dfrac{1}{3}\) - \(\dfrac{1}{2011}\))

A =  \(\dfrac{1}{2}\) \(\times\)  \(\dfrac{2008}{6033}\)

A = \(\dfrac{1004}{6033}\)

\(\dfrac{1}{3\times5}+\dfrac{1}{5\times7}+\dfrac{2}{7\times9}+..+\dfrac{1}{2009\times2011}\\ =\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{2009}-\dfrac{1}{2011}\\ =\dfrac{1}{3}-\dfrac{1}{2011}\)

Đến đây bn tự tính nhé.

\(\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+\dfrac{1}{7\cdot9}+...+\dfrac{1}{997\cdot999}\)

= \(\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{997}-\dfrac{1}{999}\right)\)

= \(\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{999}\right)\)

= \(\dfrac{1}{2}\cdot\dfrac{332}{999}=\dfrac{166}{999}\)

Đặt A = 1/3.5 + 1/5.7 + 1/7.9 + ..... + 1/99.101

=> 2A = 2/3.5 + 2/5.7 + 2/7.9 + ..... + 2/99.101

=> 2A = 1/3 - 1/5 + 1/5 - 1/7 + ..... + 1/99 - 1/101

=> 2A = 1/3 - 1/101

=> 2A = 88/303

=> A = 44/303

ai trả lời được minh

Đặt \(A=\frac{1}{1x3}+\frac{1}{3x5}+\frac{1}{5x7}+\frac{1}{7x9}\)

\(2A=\frac{2}{1x3}+\frac{2}{3x5}+\frac{2}{5x7}+\frac{2}{7x9}\)

\(2A=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{7}-\frac{1}{9}\)

\(2A=\frac{1}{1}-\frac{1}{9}=\frac{8}{9}\)

\(A=\frac{8}{9}.\frac{1}{2}=\frac{4}{9}\)

A= 1/(1x3) + 1/(3x5)+ 1/(5x7) + 1/(7x9) +  1/(9x11)

A x 2 = 2/(1x3) + 2/(3x5)+ 2/(5x7) + 2/(7x9) +  2/(9x11)

Nhận xét :

2/(1x3) = 1 - 1/3

2/(3x5) = 1/3 - 1/5

2/(5x7) = 1/5 - 1/7

2/(7x9) = 1/7 - 1/9

2/(9x11) = 1/9 - 1/11

A x 2 = 1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11

A x 2 = 1 - 1/11

A x 2 = 10/11

A = 10/11 : 2 = 5/11

các bạn k mình nha!

=1/3-1/5+1/5-1/7+1/7-1/9+...+1/43-1/45

=1/3-1/45 

1/3-1/5+1/5-1/7+1/7-1/9+...+1/43-1/45

=1/3-1/45

=14/45

Coi A=1/1x3+1/3x5+1/5x7+1/7x9

=>2A=2x(1/1x3+1/3x5+1/5x7+1/7x9)=2/1x3+2/3x5+2/5x7+2/7x9

=1-1/3+1/3-1/5+1/5-1/7+1/7-1/9

=1-1/9=8/9

=>A=8/9:2=4/9

\(\frac{1}{3x5}+\frac{1}{5x7}+\frac{1}{7x9}+\frac{1}{9x11}+\frac{1}{11x13}\)

\(=\frac{1}{2}x\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}\right)\)

\(=\frac{1}{2}x\left(\frac{1}{3}-\frac{1}{13}\right)\)

\(=\frac{1}{2}x\frac{10}{39}\)

\(=\frac{5}{39}\)

\(\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}+\frac{1}{11.13}\)

\(=\frac{1}{2}\cdot\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}\right)\)

\(=\frac{1}{2}\cdot\left(\frac{1}{3}-\frac{1}{13}\right)\)

\(=\frac{1}{2}\cdot\frac{10}{39}=\frac{5}{39}\)