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Math for Kids23 chapters | 325 lessons

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Lesson Transcript

Instructor:
*Nola Bridgens*

Nola has taught elementary school and tutored for four years. She has a bachelor's degree in Elementary Education, a master's degree in Marketing, and is a certified teacher.

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.

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.

We can put what Gauss discovered into an easy-to-use formula, which is:

**( n / 2)(first number + last number) = sum**, where

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

Take a look at this diagram to help you visually understand what the formula is saying.

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.

The smallest number is 20, and the largest number is 27.

(27 - 20) + 1 = 8.

Eight numbers make 4 pairs, and the sum of each pair is 47.

4 x 47 = 188.

The sum of the numbers from 20 to 27 is 188.

The diagram helps us see exactly what we're finding by using the formula.

Okay, let's now add the numbers 9-40. Instead of counting from 9 to 40 to see how many integers there are, we can simply subtract 9 from 40 and add 1.

(40 - 9) + 1 = 32.

There are 32 integers from 9 to 40.

Note that nine is the first number and 40 is the last number as well.

Okay, let's plug this information into our formula:

(*n* / 2)(first number + last number) = sum

(32 / 2)(9 + 40)

16 x 49 = 784

The sum of the numbers 9-40 equals 784.

By using Carl Gauss's clever formula, (*n* / 2)(first number + last number) = sum, where *n* is the number of integers, we learned how to add consecutive numbers quickly. We now know that the sum of the pairs in consecutive numbers starting with the first and last numbers is equal. We also know we can multiply the sum of their parts by the number of pairs to find the sum of the consecutive numbers. And there you have it - an easy-to-use formula for finding consecutive numbers.

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Math for Kids23 chapters | 325 lessons

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- What is an Abundant Number?
- What is a Base Number?
- What Are Cardinal Numbers? - Definition & Examples 3:06
- What Are Consecutive Numbers? - Definition & Examples 3:22
- Finding the Sum of Consecutive Numbers 4:10
- What is a Deficient Number?
- What Are Figurate Numbers? - Definition & Examples
- What Are Opposite Numbers? - Definition & Examples 2:53
- What Are Rectangular Numbers? - Definition & Examples 3:05
- Like & Unlike Terms
- Is Zero an Integer?
- Is Zero a Natural Number? 2:37
- Sieve of Eratosthenes: Lesson for Kids
- What Are Twin Prime Numbers? 3:22
- Go to Types of Numbers for Elementary School

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