Đáp án:

$\begin{array}{l}
A = \dfrac{4}{{2.5}} + \dfrac{4}{{5.8}} + \dfrac{4}{{8.11}} + ... + \dfrac{4}{{65.68}}\\
 \Rightarrow A = \dfrac{4}{3}.\left( {\dfrac{3}{{2.5}} + \dfrac{3}{{5.8}} + \dfrac{3}{{8.11}} + ... + \dfrac{3}{{65.68}}} \right)\\
 \Rightarrow A = \dfrac{4}{3}.\left( {\dfrac{1}{2} - \dfrac{1}{5} + \dfrac{1}{5} - \dfrac{1}{8} + ... + \dfrac{1}{{65}} - \dfrac{1}{{68}}} \right)\\
 \Rightarrow A = \dfrac{4}{3}.\left( {\dfrac{1}{2} - \dfrac{1}{{68}}} \right)\\
 \Rightarrow A = \dfrac{4}{3}.\dfrac{{33}}{{68}}\\
 \Rightarrow A = \dfrac{{11}}{{17}}\\
B = \dfrac{{{2^2}}}{{1.3}}.\dfrac{{{3^2}}}{{2.4}}.\dfrac{{{4^2}}}{{3.5}}...\dfrac{{{{999}^2}}}{{998.1000}}\\
 = \dfrac{{{2^2}{{.3}^2}{{.4}^2}{{...999}^2}}}{{{{1.2.3}^2}{{.4}^2}{{...998}^2}.999.1000}}\\
 = \dfrac{{2.999}}{{1000}}\\
 = \dfrac{{999}}{{500}}
\end{array}$

Đáp án:

1) `11/17`

2) `999/500`

Giải thích các bước giải:

1) `A = 4/2.5 + 4/5.8 + 4/8.11 +...+ 4/65.68`

`= 4/3 . (3/2.5 + 3/5.8 + 3/8.11+...+ 3/65.68)`

`= 4/3 . (1/2 - 1/5 + 1/5 - 1/8 + 1/8 - 1/11 +...+ 1/65 - 1/68)`

`= 4/3 . (1/2 - 1/68)`

`= 2/3 - 1/51=11/17`

2) `B=2^2/1.3. 3^2/2.4. 4^2/3.5 ... 999^2/998.1000`

`=(2.2)/1.3 . (3.3)/2.4 . (4.4)/3.5 ... (999.999)/998.1000`

`=((2.3.4...999).(2.3.4...999))/((1.2.3...998).(3.4.5...1000))=(999.2)/1000= 999/500`