S=7(1+1/3+1/6+...+1/55)
=7(2/2+2/6+...+2/110)
=14(1/2+1/6+...+1/110)
=14(1-1/2+1/2-1/3+...+1/10-11)
=14*10/11=140/11
\(S=7+\dfrac{7}{3}+\dfrac{7}{6}+\dfrac{7}{10}+\dfrac{7}{15}+\dfrac{7}{21}+\dfrac{7}{28}+\dfrac{7}{36}+\dfrac{7}{45}+\dfrac{7}{55}\\ =7+\left(\dfrac{7}{3}+\dfrac{7}{21}\right)+\left(\dfrac{7}{6}+\dfrac{7}{36}\right)+\left(\dfrac{7}{15}+\dfrac{7}{45}\right)+\dfrac{7}{10}+\dfrac{7}{28}+\dfrac{7}{55}\\ =7+\dfrac{8}{3}+\dfrac{49}{36}+\dfrac{28}{45}+\dfrac{7}{10}+\dfrac{7}{28}+\dfrac{7}{55}\\ =7+\left(\dfrac{8}{3}+\dfrac{49}{36}+\dfrac{28}{45}\right)+\dfrac{7}{10}+\dfrac{7}{28}+\dfrac{7}{55}\\ =7+\dfrac{93}{20}+\dfrac{7}{10}+\dfrac{7}{28}+\dfrac{7}{55}\\ =7+\left(\dfrac{93}{20}+\dfrac{7}{10}\right)+\dfrac{7}{28}+\dfrac{7}{55}\\ =7+\dfrac{107}{20}+\dfrac{7}{28}+\dfrac{7}{55}\\ =\dfrac{140}{11}\)
