Answer
$${S_{90}} = 8100$$
Work Step by Step
$$\eqalign{
& {\text{The first 9}}0{\text{ positive odd integers form the arithmetic sequence }} \cr
& {\text{1}},3,{\text{5}},{\text{7}},{\text{9}},{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}, \cr
& {a_1} = 1,\,\,\,\,d = 2,\,\,\,n = 90 \cr
& {\text{The formula of the sequence is}} \cr
& {a_n} = {a_1} + \left( {n - 1} \right)d \cr
& {a_{90}} = 1 + \left( {90 - 1} \right)\left( 2 \right) \cr
& {a_{90}} = 179 \cr
& \cr
& {\text{Let }}{a_1} = 1,\,\,{a_{90}} = 179,\,\,n = 90 \cr
& {\text{Using the formula }}{S_n} = \frac{n}{2}\left( {{a_1} + {a_n}} \right) \cr
& {S_{90}} = \frac{{90}}{2}\left( {1 + 179} \right) \cr
& {S_{90}} = 45\left( {180} \right) \cr
& {S_{90}} = 8100 \cr} $$