[tex]S=1+2+3+4+\cdots+996+997+998+999[/tex]
[tex]S=999+998+997+996+\cdots+4+3+2+1[/tex]
[tex]\implies 2S=(1+999)+(2+998)+\cdots+(998+2)+(999+1)[/tex]
Note that we're adding 999 terms in [tex]S[/tex], so [tex]2S[/tex] is summing together 999 pairs of numbers that add to 1000:
[tex]2S=1000+1000+\cdots+1000+1000[/tex]
[tex]2S=999(1000)[/tex]
[tex]S=\dfrac{999(1000)}2=999(500)=499500[/tex]
[tex]T=1+3+5+\cdots+993+995+997[/tex]
[tex]T=997+995+993+\cdots+5+3+1[/tex]
[tex]\implies 2T=(1+997)+(3+995)+\cdots+(995+3)+(997+1)[/tex]
This time, [tex]T[/tex] adds up 499 numbers - we can determine this by finding the value of [tex]n[/tex] such that [tex]2n-1=997[/tex] - that each add up to 998, so
[tex]2T=998+998+\cdots+998+998[/tex]
[tex]2T=499(998)[/tex]
[tex]T=\dfrac{499(998)}2=499(499)=249001[/tex]
