Question

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

O=3+5+7+9+...+105
E=2+4+6+8+...+104

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`.