Bài giải:

$A=\frac{5}{6}+\frac{5}{66}+\frac{5}{176}+\frac{5}{336}+\frac{5}{546}+\frac{5}{806}$

$=\frac{5}{1\times 6}+\frac{5}{6\times 11}+\frac{5}{11\times 16}+\frac{5}{16\times 21}+\frac{5}{21\times 26}+\frac{5}{26\times 31}$

$=1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+\frac{1}{16}-\frac{1}{21}+\frac{1}{21}-\frac{1}{26}+\frac{1}{26}-\frac{1}{31}$

$=1-\frac{1}{31}$

$=\frac{30}{31}$