Bài giải:

$A=\frac{4}{21}+\frac{4}{77}+\frac{4}{165}+\frac{4}{285}+\frac{4}{437}+\frac{4}{621}$

$A=\frac{4}{3\times 7}+\frac{4}{7\times 11}+\frac{4}{11\times 15}+\frac{4}{15\times 19}+\frac{4}{19\times 23}+\frac{4}{23\times 27}$

$=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+\frac{1}{15}-\frac{1}{19}+\frac{1}{19}-\frac{1}{23}+\frac{1}{23}-\frac{1}{27}$

$=\frac{1}{3}-\frac{1}{27}$

$=\frac{8}{27}$