$A = \dfrac{7}{10.11}+\dfrac{7}{11.12}+...+\dfrac{7}{69.70} $
$=7.( \dfrac{1}{10.11}+\dfrac{1}{11.12}+...+\dfrac{1}{69.70} ) $
$= 7.(\dfrac{1}{10} - \dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{12}+....+\dfrac{1}{69}-\dfrac{1}{70}) $
$=7.(\dfrac{1}{10} - \dfrac{1}{70}) $
$=7.\dfrac{3}{35} = \dfrac{3}{5}$