Answer
$S_{1}=\dfrac {1}{6};$
$S_{2}=S_{1}+\dfrac {1}{6}=\dfrac {1}{3};$
$S_{3}=S_{2}+\dfrac {3}{20}=\dfrac {29}{60}$
$S_{4}=S_{3}+\dfrac {2}{15}=\dfrac {37}{60};$
$S_{5}=S_{4}+\dfrac {5}{42}=\dfrac {103}{140}$
Work Step by Step
$S_{n}=\dfrac {1}{2\times 3}+\dfrac {2}{3\times 4}+\dfrac {3}{4\times 5}+\dfrac {4}{5\times 6}+\dfrac {5}{6\times 7}+\ldots \dfrac {n}{\left( n+1\right) \left( n+2\right) }\Rightarrow $
$S_{1}=\dfrac {1}{6};$
$S_{2}=S_{1}+\dfrac {1}{6}=\dfrac {1}{3};$
$S_{3}=S_{2}+\dfrac {3}{20}=\dfrac {29}{60}$
$S_{4}=S_{3}+\dfrac {2}{15}=\dfrac {37}{60};$
$S_{5}=S_{4}+\dfrac {5}{42}=\dfrac {103}{140}$
