$\text{$\dfrac{1}{6}$ + $\dfrac{1}{12}$ + $\dfrac{1}{20}$ + $\dfrac{1}{30}$ + $\dfrac{1}{42}$ + $\dfrac{1}{56}$}$
$\text{= $\dfrac{3-2}{6}$ + $\dfrac{4-3}{12}$ + $\dfrac{5-4}{20}$ + $\dfrac{6-5}{30}$ + $\dfrac{7-6}{42}$ + $\dfrac{8-7}{56}$}$
$\text{= $\dfrac{1}{2}$ - $\dfrac{1}{3}$ + $\dfrac{1}{3}$ - $\dfrac{1}{4}$ + $\dfrac{1}{4}$ - $\dfrac{1}{5}$ + $\dfrac{1}{5}$ - $\dfrac{1}{6}$ + $\dfrac{1}{6}$ - $\dfrac{1}{7}$ + $\dfrac{1}{7}$ - $\dfrac{1}{8}$}$
$\text{= $\dfrac{1}{2}$ - $\dfrac{1}{8}$}$
$\text{= $\dfrac{3}{8}$}$
