Question

Question #e35c4

Answer 1

`2500`

Explanation:

`1+3+5+...+97+99`

Notice the following: the first and last term add to be:
`1+99=100`

The second and second-to-last:
`3+97=100`

Third and third-to-last:
`5+95=100`

This pattern continues until `49+51=100`

Now, all we need to do is figure out how many odd numbers there are from `1` to `49`, inclusive.

Well there are `5` odd numbers in every multiple of `10` (`1,3,5,7,9` or `11,13,15,17,19`). Since there are `5` multiples of `10` between `1` and `49` (the digits, the tens, the twenties, the thirties, and the forties), there are `5*5=25` odd numbers.

`25*100=2500`.

Just as a side note, it can be shown that the sum of the odd numbers from `1` to `2n+1` is `n^2`.