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".*

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