Answer
$${S_1} = \frac{1}{2},{S_2} = \frac{5}{6},{S_3} = \frac{{13}}{{12}},{S_4} = \frac{{77}}{{60}},{S_5} = \frac{{29}}{{20}}$$
Work Step by Step
$$\eqalign{
& {a_n} = \frac{1}{{n + 1}} \cr
& {\text{Find the first five terms of the sequence}} \cr
& {a_1} = \frac{1}{2} \cr
& {a_2} = \frac{1}{3} \cr
& {a_3} = \frac{1}{4} \cr
& {a_4} = \frac{1}{5} \cr
& {a_5} = \frac{1}{6} \cr
& {\text{Then by definition of partial sum}} \cr
& {S_1} = {a_1} \cr
& {S_1} = \frac{1}{2} \cr
& {S_2} = {a_1} + {a_2} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \cr
& {S_3} = {a_1} + {a_2} + {a_3} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{{13}}{{12}} \cr
& {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{{77}}{{60}} \cr
& {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} = \frac{{29}}{{20}} \cr
& {\text{the partial sums are}}: \cr
& {S_1} = \frac{1}{2},{S_2} = \frac{5}{6},{S_3} = \frac{{13}}{{12}},{S_4} = \frac{{77}}{{60}},{S_5} = \frac{{29}}{{20}} \cr} $$
