Đáp án:

$\begin{array}{l}
A = \dfrac{2}{{3.7}} + \dfrac{3}{{7.13}} + \dfrac{4}{{13.21}} + \dfrac{5}{{21.31}} + \dfrac{6}{{31.43}} + \dfrac{7}{{43.57}}\\
 \Leftrightarrow 2.A = \dfrac{4}{{3.7}} + \dfrac{6}{{7.13}} + \dfrac{8}{{13.21}} + \dfrac{{10}}{{21.31}} + \dfrac{{12}}{{31.43}} + \dfrac{{14}}{{43.57}}\\
 = \dfrac{{7 - 3}}{{3.7}} + \dfrac{{13 - 7}}{{7.13}} + \dfrac{{21 - 13}}{{13.21}} + \dfrac{{31 - 21}}{{21.31}} + \dfrac{{43 - 31}}{{31.43}}\\
 + \dfrac{{57 - 43}}{{43.57}}\\
 = \dfrac{1}{3} - \dfrac{1}{7} + \dfrac{1}{7} - \dfrac{1}{{13}} + \dfrac{1}{{13}} - \dfrac{1}{{21}} + \dfrac{1}{{21}} - \dfrac{1}{{31}}\\
 + \dfrac{1}{{31}} - \dfrac{1}{{43}} + \dfrac{1}{{43}} - \dfrac{1}{{57}}\\
 = \dfrac{1}{3} - \dfrac{1}{{57}}\\
 = \dfrac{{19 - 1}}{{57}}\\
 = \dfrac{{18}}{{57}}\\
 \Leftrightarrow A = \dfrac{{18}}{{57}}:2\\
 \Leftrightarrow A = \dfrac{9}{{57}} = \dfrac{3}{{19}}
\end{array}$

Sửa đề: Vì nếu đề thế kia thì sẽ không có quy luật để tính nhanh.

`A=2/(3xx7)+3/(7xx13)+4/(13xx21)+5/(21xx31)+6/(31xx43)+7/(43xx57)`

`Axx2=4/(3xx7)+6/(7xx13)+8/(13xx21)+10/(21xx31)+12/(31xx43)+14/(43xx57)`

`Axx2=1/3-1/7+1/7-1/13+1/13-1/21+1/21-1/31+1/31-1/43+1/43-1/57`

`Axx2=1/3-1/57`

`Axx2=1/57`

`A=1/57:2`

`A=3/19`

Vậy `A=3/19`