\(1-\dfrac{5}{14}-\dfrac{5}{84}-\dfrac{5}{204}-\dfrac{5}{374}\) =\(1-\left(\dfrac{1}{1.14}+\dfrac{1}{14.6}+\dfrac{1}{6.34}+\dfrac{1}{34.11}\right)\)
= \(1-\left(\dfrac{1}{1}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{34}+\dfrac{1}{34}-\dfrac{1}{11}\right)\)
= 1 - \(\left(1-\dfrac{1}{11}\right)\) = \(1-\dfrac{10}{11}=\dfrac{1}{11}\)
\(=1-\dfrac{5}{2\cdot7}-\dfrac{5}{7\cdot12}-\dfrac{5}{12\cdot17}-\dfrac{5}{17\cdot22}\)
\(=1-\left(\dfrac{1}{2}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{12}+\dfrac{1}{12}-\dfrac{1}{17}+\dfrac{1}{17}-\dfrac{1}{22}\right)\)
\(=5-\dfrac{11-1}{22}=5-\dfrac{10}{22}=5-\dfrac{5}{11}=\dfrac{50}{11}\)
