Answer
$${S_1} = \frac{1}{{12}},{S_2} = \frac{{37}}{{100}},{S_3} = \frac{{103}}{{700}},{S_4} = \frac{{1027}}{{6300}},{S_5} = \frac{{24,169}}{{138,600}}$$
Work Step by Step
$$\eqalign{
& {a_n} = \frac{1}{{\left( {n + 3} \right)\left( {2n + 1} \right)}} \cr
& {\text{Find the first five terms of the sequence}} \cr
& {a_1} = \frac{1}{{\left( {1 + 3} \right)\left( {2\left( 1 \right) + 1} \right)}} = \frac{1}{{12}} \cr
& {a_2} = \frac{1}{{\left( {2 + 3} \right)\left( {2\left( 2 \right) + 1} \right)}} = \frac{1}{{25}} \cr
& {a_3} = \frac{1}{{\left( {3 + 3} \right)\left( {2\left( 3 \right) + 1} \right)}} = \frac{1}{{42}} \cr
& {a_4} = \frac{1}{{\left( {4 + 3} \right)\left( {2\left( 4 \right) + 1} \right)}} = \frac{1}{{63}} \cr
& {a_5} = \frac{1}{{\left( {5 + 3} \right)\left( {2\left( 5 \right) + 1} \right)}} = \frac{1}{{88}} \cr
& {\text{Then by definition of partial sum}} \cr
& {S_1} = {a_1} = \frac{1}{{12}} \cr
& {S_2} = {a_1} + {a_2} = \frac{1}{{12}} + \frac{1}{{25}} = \frac{{37}}{{100}} \cr
& {S_3} = {a_1} + {a_2} + {a_3} = \frac{1}{{12}} + \frac{1}{{25}} + \frac{1}{{42}} = \frac{{103}}{{700}} \cr
& {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{{12}} + \frac{1}{{25}} + \frac{1}{{42}} + \frac{1}{{63}} = \frac{{1027}}{{6300}} \cr
& {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = \frac{1}{{12}} + \frac{1}{{25}} + \frac{1}{{42}} + \frac{1}{{63}} + \frac{1}{{88}} = \frac{{24,169}}{{138,600}} \cr
& {\text{the partial sums are}}: \cr
& {S_1} = \frac{1}{{12}},{S_2} = \frac{{37}}{{100}},{S_3} = \frac{{103}}{{700}},{S_4} = \frac{{1027}}{{6300}},{S_5} = \frac{{24,169}}{{138,600}} \cr} $$
