$A = \dfrac{7}{10.11} + \dfrac{7}{11.12}+\dfrac{7}{12.13} +...+\dfrac{7}{69.70}$
$=7.(\dfrac{1}{10.11} + \dfrac{1}{11.12}+\dfrac{1}{12.13} +...+\dfrac{1}{69.70})$
$=7.(\dfrac{1}{10} - \dfrac{1}{11} + \dfrac{1}{11} -\dfrac{1}{12}+\dfrac{1}{12}-\dfrac{1}{13} + ...+\dfrac{1}{69} - \dfrac{1}{70})$
$=7.(\dfrac{1}{10} - \dfrac{1}{70})$
$=7.\dfrac{7 - 1}{70}$
$=7.\dfrac{3}{35}$
$=\dfrac{7.3}{35}$
$=\dfrac{3}{5}$
