Answer
$${S_{50}} = 2500$$
Work Step by Step
$$\eqalign{
& {\text{The first }}50{\text{ positive odd integers form the arithmetic sequence }} \cr
& {\text{1}},3,{\text{5}},{\text{7}},{\text{9}},{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}, \cr
& {a_1} = 1,\,\,\,\,d = 2,\,\,\,n = 50 \cr
& {\text{The formula of the sequence is}} \cr
& {a_n} = {a_1} + \left( {n - 1} \right)d \cr
& {a_{50}} = 1 + \left( {50 - 1} \right)\left( 2 \right) \cr
& {a_{50}} = 99 \cr
& \cr
& {\text{Let }}{a_1} = 1,\,\,{a_{50}} = 99,\,\,n = 50 \cr
& {\text{Using the formula }}{S_n} = \frac{n}{2}\left( {{a_1} + {a_n}} \right) \cr
& {S_{50}} = \frac{{50}}{2}\left( {1 + 99} \right) \cr
& {S_{50}} = 25\left( {100} \right) \cr
& {S_{50}} = 2500 \cr} $$
