Back To Course

Math for Kids23 chapters | 325 lessons

{{courseNav.course.topics.length}} chapters | {{courseNav.course.mDynamicIntFields.lessonCount}} lessons | {{course.flashcardSetCount}} flashcard set{{course.flashcardSetCoun > 1 ? 's' : ''}}

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 79,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Log in here for access

Your next lesson will play in
10 seconds

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.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Log in here for access

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
5 in chapter 12 of the course:

Back To Course

Math for Kids23 chapters | 325 lessons

{{courseNav.course.topics.length}} chapters | {{courseNav.course.mDynamicIntFields.lessonCount}} lessons | {{course.flashcardSetCount}} flashcard set{{course.flashcardSetCoun > 1 ? 's' : ''}}

- 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? 2:17
- 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

- Humanities 201: Critical Thinking & Analysis
- SH Hotels Mentorship Program
- SH Hotels Leadership Development Program Part 3 - Environmental Science & Sustainability Certificate
- SH Hotels Leadership Development Program Part 2 - Leadership Certificate
- SH Hotels Leadership Development Program Part 1 - Hospitality & Tourism Management Certificate
- CEOE English: American Literature in the 1800's
- CEOE English: English Literature in the 20th Century
- CEOE English: Types of Poetry
- Defining Critical Thinking
- Logical Fallacies & Critical Thinking
- What is the Series 65 License Exam?
- How to Pass the SIE License Exam
- What is the Series 63 License Exam?
- How to Pass a Finance Certification Exam
- What is the Series 7 License Exam?
- How to Get an SIE License
- How to Get a Series 7 License

- Modernism in Architecture: Definition & History
- Compound vs. Complex Sentences in English
- The Boarded Window Summary
- Symmetric Encryption: Definition & Example
- Real Estate Broker Responsibilities in Mississippi
- The Alienist: Structure & Style
- Arkansas Real Estate Regulations for Agency Relationships
- Quiz & Worksheet - Hatchet Chapter 8
- Quiz & Worksheet - Questions for Freak the Mighty Chapter 11
- Quiz & Worksheet - Questions for Freak the Mighty Chapter 10
- Quiz & Worksheet - Balance in Graphic Design
- Flashcards - Real Estate Marketing Basics
- Flashcards - Promotional Marketing in Real Estate
- Grammar Games
- Teaching Strategies | Instructional Strategies & Resources

- Sales and Marketing: Help & Review
- All Quiet on the Western Front Study Guide
- WEST Elementary Education Subtest II (103): Practice & Study Guide
- Building a Customer Service Team
- Ohio State Test - Physical Science: Practice & Study Guide
- Philosophy and Nonfiction: Help and Review
- Sentence Structure: Understanding Grammar: Help and Review
- Quiz & Worksheet - Reverend Parris Quotes
- Quiz & Worksheet - Resource Depletion
- Quiz & Worksheet - Questions for A Wrinkle in Time Chapter 5
- Quiz & Worksheet - Strategic Communication Imperative
- Quiz & Worksheet - Message Follow-Up in Business Communication

- Absorption Costing: Income Statement & Marginal Costing
- Thermochemical Equations
- Reading Comprehension Activities
- About the GED Mathematical Reasoning Test
- ELM Test Dates
- WIDA Can Do Descriptors for Grades 9-12
- Sounds Experiments Lesson for Kids
- Map Activities for Kids
- What is the MCAT?
- Who is Study.com?
- ELM Test Dates
- How to Get a Job as a Teacher

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

- Show that \sum_{k=1}^n k = \frac{n(n+1)}{2}
- Compute the sum of (1)(2)+(2)(3)+(3)(4)......(100)(101)
- Calculate the following sum: \sum_{j=1}^{m} \sum_{k=1}^{m} jk According to WolframAlpha the answer is : \frac{[m^2(m+1)^2] }{4}
- Evaluate the sum. \left ( \sum_{k=1}^{7} k\right )^{2}-\sum_{k=1}^{7}\frac{k^{3}}{4}
- How many different rectangles can be drawn on an 8 \times 8 chessboard? Each rectangle can have height and width (which may differ) of 1 through 8 squares, and two rectangles are different if the su
- The sum of three consecutive even integers is 108. What is the largest number?
- What do you get when you add up all the numbers from 1 to 100 consecutively?
- What is the sum of all of the odd numbers from 1 to 59?
- Use the summation formulas to rewrite \displaystyle \sum_{i=1}^{n}\frac{3i+2}{n^2} without the summation notation.
- Find the sum of the first 230 positive even whole numbers.

Browse by subject