[tex]S=1+3+5+\cdots+995+997+999[/tex]
[tex]S=999+997+995+\cdots+5+3+1[/tex]
[tex]\implies 2S=(1+999)+(3+997)+(5+995)+\cdots+(995+5)+(997+3)+(999+1)[/tex]
[tex]2S=\underbrace{1000+1000+\cdots+1000+1000}_{500\text{ times}}[/tex]
(We are adding 500 instances of 1000 because that's how many odd numbers there are between 1 and 1000, inclusive.)
[tex]2S=500\times1000=500000[/tex]
[tex]\implies S=250000[/tex]
