How to Add Numbers from 1 to 100

How fast could you find the sum of the numbers from 1 to 100? There is a famous story that a German school teacher once asked his class to work on this, thinking it would keep them busy. One of his students was future physicist Friedrich Gauss, who quickly figured out a trick to do the calculation without any addition. Here is a method for adding the numbers 1 to 100, based on Gauss's trick.

Difficulty: Moderate

Instructions

1. First clarify the problem. You are adding 1 + 2 + 3 + 4 + 5...+ 96 + 97 + 98 + 99 + 100. If you were adding some portion that began with a number other than 1, or if you were adding just odds or evens, the method would be different.

2. Notice that the first and last numbers add up to 101 (1 + 100). So do the second and second-to-last (2 + 99). In fact, all of the numbers pair up like this, always adding up to 101.

3. If you paired up the numbers this way, you'd have 50 pairs, which is half of 100. Each pair's sum is 101.

4. The formula that summarizes this trick uses "n" to stand for the top number, in this case 100. N multiplied by (n + 1), divided by 2, gives you the answer. (100 x 101) / 2 is the same as 50 x 101. 50 x 101 = 5050.

5. The formula also works for higher sequences, as long as you are adding all the counting numbers, starting with (and including) 1.