Odd numbers take the form [tex]2n-1[/tex], where [tex]n\ge1[/tex] is an integer. When [tex]n=400[/tex], the last odd number would be 799. So we're adding
[tex]S=1+3+5+\cdots+795+797+799[/tex]
By reversing the order of terms, we have
[tex]S=799+797+795+\cdots+5+3+1[/tex]
and we can pair up terms in both sums at the same position to write
[tex]2S=(1+799)+(3+797)+(5+795)+\cdots(795+5)+(797+3)+(799+1)[/tex]
so that we are basically adding 400 copies of 800, and from there we can find the value of the sum right away:
[tex]2S=400\cdot800\implies S=160,000[/tex]
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We could also make use of the formulas,
[tex]\displaystyle\sum_{i=1}^n1=n[/tex]
[tex]\displaystyle\sum_{i=1}^ni=\dfrac{n(n+1)}2[/tex]
We have
[tex]S=\displaystyle\sum_{i=1}^{400}(2i-1)=2\sum_{i=1}^{400}i-\sum_{i=1}^{400}1=400(400+1)-400=400^2=160,000[/tex]
