Answer
$${S_1} = \frac{1}{{12}},{S_2} = \frac{2}{{15}},{S_3} = \frac{1}{6},{S_4} = \frac{4}{{21}},{S_5} = \frac{5}{{24}}$$
Work Step by Step
$$\eqalign{
& {a_n} = \frac{1}{{\left( {n + 2} \right)\left( {n + 3} \right)}} \cr
& {\text{Find the first five terms of the sequence}} \cr
& {a_1} = \frac{1}{{\left( {1 + 2} \right)\left( {1 + 3} \right)}} = \frac{1}{{12}} \cr
& {a_2} = \frac{1}{{\left( {2 + 2} \right)\left( {2 + 3} \right)}} = \frac{1}{{20}} \cr
& {a_3} = \frac{1}{{\left( {3 + 2} \right)\left( {3 + 3} \right)}} = \frac{1}{{30}} \cr
& {a_4} = \frac{1}{{\left( {4 + 2} \right)\left( {4 + 3} \right)}} = \frac{1}{{42}} \cr
& {a_5} = \frac{1}{{\left( {5 + 2} \right)\left( {5 + 3} \right)}} = \frac{1}{{56}} \cr
& {\text{Then by definition of partial sum}} \cr
& {S_1} = {a_1} = \frac{1}{{12}} \cr
& {S_2} = {a_1} + {a_2} = \frac{1}{{12}} + \frac{1}{{20}} = \frac{2}{{15}} \cr
& {S_3} = {a_1} + {a_2} + {a_3} = \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} = \frac{1}{6} \cr
& {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} = \frac{4}{{21}} \cr
& {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} = \frac{5}{{24}} \cr
& {\text{the partial sums are}}: \cr
& {S_1} = \frac{1}{{12}},{S_2} = \frac{2}{{15}},{S_3} = \frac{1}{6},{S_4} = \frac{4}{{21}},{S_5} = \frac{5}{{24}} \cr} $$