Answer
$${S_1} = \frac{1}{7},{S_2} = \frac{{16}}{{63}},{S_3} = \frac{{239}}{{693}},{S_4} = \frac{{3800}}{{9009}},{S_5} = \frac{{22,003}}{{45,045}}$$
Work Step by Step
$$\eqalign{
& {a_n} = \frac{1}{{2n + 5}} \cr
& {\text{Find the first five terms of the sequence}} \cr
& {a_1} = \frac{1}{{2\left( 1 \right) + 5}} = \frac{1}{7} \cr
& {a_2} = \frac{1}{{2\left( 2 \right) + 5}} = \frac{1}{9} \cr
& {a_3} = \frac{1}{{2\left( 3 \right) + 5}} = \frac{1}{{11}} \cr
& {a_4} = \frac{1}{{2\left( 4 \right) + 5}} = \frac{1}{{13}} \cr
& {a_5} = \frac{1}{{2\left( 5 \right) + 5}} = \frac{1}{{15}} \cr
& {\text{Then by definition of partial sum}} \cr
& {S_1} = {a_1} = \frac{1}{7} \cr
& {S_2} = {a_1} + {a_2} = \frac{1}{7} + \frac{1}{9} = \frac{{16}}{{63}} \cr
& {S_3} = {a_1} + {a_2} + {a_3} = \frac{1}{7} + \frac{1}{9} + \frac{1}{{11}} = \frac{{239}}{{693}} \cr
& {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{7} + \frac{1}{9} + \frac{1}{{11}} + \frac{1}{{13}} = \frac{{3800}}{{9009}} \cr
& {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = \frac{1}{7} + \frac{1}{9} + \frac{1}{{11}} + \frac{1}{{13}} + \frac{1}{{15}} = \frac{{22,003}}{{45,045}} \cr
& {\text{the partial sums are}}: \cr
& {S_1} = \frac{1}{7},{S_2} = \frac{{16}}{{63}},{S_3} = \frac{{239}}{{693}},{S_4} = \frac{{3800}}{{9009}},{S_5} = \frac{{22,003}}{{45,045}} \cr} $$
