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How long would it take you to add the numbers 1-100? Without the quick thinking of Carl Gauss, it might take you a long time. In this lesson, we’ll learn the formula Gauss discovered to add consecutive numbers and how to apply it.
Gauss & Consecutive Numbers
There once was a little boy named Carl Gauss. He had a very lazy teacher who didn’t want to teach one morning, so the teacher gave the class an assignment to add the numbers 1-100. The teacher thought for sure this would take the class a while, and he could take a short nap. To his surprise, Carl came up with the answer (5,050) in about a minute. The teacher thought Carl had cheated and asked him to explain how he had come up with his answer so quickly.
Carl noticed very quickly that the sum was the same when he added the first and last number, the second and second-to-last number, the third and third-to-last number, and so on. He figured out that because there were 100 numbers, there would be 50 pairs that were equal to 101. The sum of the numbers 1-100 would be equal to the number of pairs (50) multiplied by the sum of each pair (101), or 50 x 101 = 5,050. Karl was able to use what he knew about numbers to solve what seemed like a complicated assignment in a short amount of time.
Using the Formula
We can put what Gauss discovered into an easy-to-use formula, which is:
(n / 2)(first number + last number) = sum, where n is the number of integers.
Let’s use the example of adding the numbers 1-100 to see how the formula works.
Find the sum of the consecutive numbers 1-100:
(100 / 2)(1 + 100)
50(101) = 5,050
More Examples
Take a look at this diagram to help you visually understand what the formula is saying.
consecutive sum
Let’s use the formula to add the numbers 20-27. We know there are a total of 8 numbers from 20-27. In this example, we can count by looking at the diagram, but we can also find the total number of integers by subtracting the smallest number from the largest number and adding 1.
 
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