A = \(\dfrac{1}{2^2}\) + \(\dfrac{1}{4^2}\) + \(\dfrac{1}{6^2}\) + .....+ \(\dfrac{1}{1002^2}\)

A = \(\dfrac{1}{2^2.1^2}\) + \(\dfrac{1}{2^2.2^2}\) + \(\dfrac{1}{2^2.3^2}\)+......+\(\dfrac{1}{2^2.501^2}\)

A = \(\dfrac{1}{2^2}\) \(\times\)( \(1\) + \(\dfrac{1}{2^2}\) + \(\dfrac{1}{3^2}\)+.......+ \(\dfrac{1}{501^2}\))

ta có : \(\dfrac{1}{2^2}\)   < \(\dfrac{1}{1.2}\)

           \(\dfrac{1}{3^2}\)   < \(\dfrac{1}{2.3}\)

          ................

         \(\dfrac{1}{501^2}\) < \(\dfrac{1}{500.501}\)

Cộng vế với vế ta được

           \(\dfrac{1}{2^2}\) + \(\dfrac{1}{3^2}\) +.....+ \(\dfrac{1}{501^2}\) < \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+.......+\dfrac{1}{500.501}\)

           \(\dfrac{1}{2^2}\) +  \(\dfrac{1}{3^2}\) +.....+ \(\dfrac{1}{501^2}\) < \(\dfrac{1}{1}\) - \(\dfrac{1}{2}\) + \(\dfrac{1}{2}-\dfrac{1}{3}\)+.....+ \(\dfrac{1}{500}-\dfrac{1}{501}\)

            \(\dfrac{1}{2^2}\) + \(\dfrac{1}{3^2}\)+......+ \(\dfrac{1}{501^2}\) < 1 - \(\dfrac{1}{501}\) < 1 

   =>A = \(\dfrac{1}{4}\) \(\times\) ( 1 + \(\dfrac{1}{2^2}\)+ \(\dfrac{1}{3^2}\)+.....+\(\dfrac{1}{501^2}\)) < \(\dfrac{1}{4}\) \(\times\)(1 + 1)

    A <  \(\dfrac{1}{4}\)  \(\times\) 2

    A < \(\dfrac{1}{2}\)

vì phải quy đồng thì mới ra được mẫu số là 10

suy ra 2x/10-1/2y=1/10

vì 2x-1=1 suy ra 2x =2 vạy x =1

vì mẫu số trừ mẫu số thì giữ nguyên suy ra 2y=10 vậy y=5

10,98751=10,9=10,99=10,998

0,09877=0,1=0,99=0,999

0,[45]=0,5=0,44=0,455

123,4567=123,5=123,46=123,457

`A =2/15 +2/35 +2/63 +... +2/339`

`= 2/(3.5) +2/(5.7) + 2/(7.9) + ...+2/(19.21)`

`= 1/3 -1/5 +1/5 -1/7 +1/7 -1/9 +... 1/19 -1/21`

`= 1/3 -1/21 = 7/21 -1/21`

`=6/21 = 2/7`

$A=2/15+2/35+2/63+...+2/339$

$=2/(3.5)+2/(5.7)+2/(7.9)+...+2/(19.21)$

$=1/3-1/5+1/5-1/7+1/7-1/9+...1/19-1/21$

$=1/3-1/21=7/21-1/21$

$=6/21=2/7$

Lời giải:

$\Rightarrow A-B=-1$

`-3/3*15/13-3/7*11/13-3/7`

`-1*15/13-3/7*11/13-3/7`

`=-3/7:3/7*15/13-3/7*11/13-3/7`

`=-3/7*35/13-3/7*11/13-3/7`

`=-(3/7*35/13+3/7*11/13+3/7)`

`=-3/7(35/13+11/13+1)`

`=-3/7*59/13`

`=-177/91`

$-3/3\ast 15/13-3/7\ast 11/13-3/7$

$-1\ast 15/13-3/7\ast 11/13-3/7$

$=-3/7:3/7\ast 15/13-3/7\ast 11/13-3/7$

$=-3/7\ast 35/13-3/7\ast 11/13-3/7$

$=-(3/7\ast 35/13+3/7\ast 11/13+3/7)$

$=-3/7(35/13+11/13+1)$

$=-3/7\ast 59/13$

$=-177/91$

`1/2+3/5(x-2)=1/5`

`=>3/5(x-2)=1/5-1/2`

`=>3/5(x-2)=2/10 -5/10`

`=>3/5(x-2)=-3/10`

`=> x-2= -3/10 : 3/5`

`=> x-2= -3/10 xx 5/3`

`=> x-2=-1/2`

`=>x=-1/2 +2`

`=>x=-1/2 + 4/2`

`=>x= 3/2`

$1/2+3/5(x-2)=1/5$

$=>3/5(x-2)=1/5-1/2$

$=>3/5(x-2)=2/10-5/10$

$=>3/5(x-2)=-3/10$

$=>x-2=-3/10:3/5$

$=>x-2=-3/10xx5/3$

$=>x-2=-1/2$

$=>x=-1/2+2$

$=>x=-1/2+4/2$

$=>x=3/2$

\(A=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+.....+\dfrac{1}{2021.2023}\)

\(=\dfrac{1}{2}.\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+....+\dfrac{2}{2021.2023}\right)\)

\(=\dfrac{1}{2}.\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+....+\dfrac{1}{2021}-\dfrac{1}{2023}\right)\)

\(=\dfrac{1}{2}.\left(1-\dfrac{1}{2023}\right)=\dfrac{1}{2}.\dfrac{2022}{2023}=\dfrac{1011}{2023}\)

Ta có A = \(\dfrac{1}{1\cdot3}+\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+...+\dfrac{1}{2021\cdot2023}\)

            = \(\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{2021\cdot2023}\right)\)

            = \(\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{2021}+\dfrac{1}{2023}\right)\)

            = \(\dfrac{1}{2}\left(1-\dfrac{1}{2023}\right)=\dfrac{1}{2}\cdot\dfrac{2022}{2023}=\dfrac{1011}{2023}\)