Answer
$$10000$$
Work Step by Step
$$\eqalign{
& \sum\limits_{k = 1}^{100} {\left( {2k - 1} \right)} \cr
& {\text{The first term is}} \cr
& {a_1} = 2\left( 1 \right) - 1 = 1 \cr
& {\text{The last term is}} \cr
& {a_{100}} = 2\left( {100} \right) - 1 = 199 \cr
& {\text{Using the formula }}{S_n} = \frac{n}{2}\left( {{a_1} + {a_n}} \right),{\text{ we obtain}} \cr
& \sum\limits_{k = 1}^{100} {\left( {2k - 1} \right)} = {S_{100}} = \frac{{100}}{2}\left( {1 + 199} \right) \cr
& \sum\limits_{k = 1}^{100} {\left( {2k - 1} \right)} = 10000 \cr} $$
