Question

Without computing each sum, find which is greater, O or E, and by how much.

Answer 1

$O=3+5+7+9+\cdots+105$
$O=(2+1)+(4+1)+(6+1)+(8+1)+\cdots+(104+1)$
$O=(2+4+6+8+\cdots+104)+n$

where $n$ is the numbers of 1s, which is the same as the number of terms being summed in either $O$ or $E$. It's easy to see what that number is from $E$; factoring a 2 from each term, you end up adding up the consecutive integers between 1 and 52, so $n=52$.

So $O>E$, and $O=E+52\implies O-E=52$.