Computing the Sum of All Numbers from F to L

If you want to add all the numbers from F (First) to L (Last), here is an easy way to do it, and many times can be performed in your head.

$\sum_{F}^{L} = (L^{2} - F^{2} + F + L)/2$

Where F is the first number in the number line and L is the Last.

Example:

To add all the numbers from 1 to 10, plug in 1 for F and 10 for L.

$$(10^{2} - 1^{2} + 10 + 1)/2 =$$

$$(100 - 1 + 11)/2 =$$

$$110/2 = 55$$

The result is the same as adding all the numbers from 1 to 10 like this:

$$1+2+3+4+5+6+7+8+9+10=55$$

This works for all nonnegative integers where 0 < F < L. You don't have to start with 1.

Submitted by former curiousmath.com member "deud".