Skip to Content

Log in / Sign up

wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow

How to Add Consecutive Integers from 1 to 100

Download Article

Reviewed by Joseph Meyer

Last Updated: September 30, 2022 Fact Checked

Download Article

According to math legend, the mathematician Carl Friedrich Gauss, at the age of 8, came up with a method for quickly adding the consecutive numbers between 1 and 100. ^{[1]
X
Research source} The basic method is pairing numbers in the group, then multiplying the sum of each pair by the number of pairs. From this method we can derive a formula for adding consecutive numbers $a_{{1}}$ through $a_{{n}}$ : $S_{{n}}=n({\frac {a_{{1}}+a_{{n}}}{2}})$ . These methods can be applied to any series of consecutive numbers, not just 1 through 100.

Method 1

Method 1 of 2:

Using the Formula for the Sum of a Series

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/0\/03\/Add-Consecutive-Integers-from-1-to-100-Step-1-Version-2.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/0\/03\/Add-Consecutive-Integers-from-1-to-100-Step-1-Version-2.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

The formula is $S_{{n}}=n({\frac {a_{{1}}+a_{{n}}}{2}})$ , where $n$ equals the number of terms in the series, $a_{{1}}$ is the first number in the series, $a_{{n}}$ is the last number in the series, and $S_{{n}}$ equals the sum of $n$ numbers. ^{[2]
X
Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/2a\/Add-Consecutive-Integers-from-1-to-100-Step-2-Version-2.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-2-Version-2.jpg","bigUrl":"\/images\/thumb\/2\/2a\/Add-Consecutive-Integers-from-1-to-100-Step-2-Version-2.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-2-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

2
Plug the values into the formula. This means substituting the first term in the series for $a_{{1}}$ , and the last term in the series for $a_{{n}}$ . When adding consecutive numbers 1 through 100, $a_{{1}}=1$ and $a_{{n}}=100$ .
- Thus, your formula will look like this: $S_{{100}}=n({\frac {1+100}{2}})$ .
Advertisement
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/57\/Add-Consecutive-Integers-from-1-to-100-Step-3-Version-2.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-3-Version-2.jpg","bigUrl":"\/images\/thumb\/5\/57\/Add-Consecutive-Integers-from-1-to-100-Step-3-Version-2.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-3-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

Since $100+1=101$ , you will divide 101 by 2: ${\frac {101}{2}}=50.5$ .
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b1\/Add-Consecutive-Integers-from-1-to-100-Step-4-Version-2.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-4-Version-2.jpg","bigUrl":"\/images\/thumb\/b\/b1\/Add-Consecutive-Integers-from-1-to-100-Step-4-Version-2.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-4-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

4
Multiply by $n$ . This will give you the sum the consecutive number in the series. In this instance, since you are adding consecutive numbers to 100, $n=100$ . So, you would calculate $100(50.5)=5050$ . Thus, the sum of the consecutive numbers between 1 and 100 is 5,050. ^{[3]
X
Research source}
- To quickly multiply a number by 100, move the decimal point two places to the right. ^{[4]
  X
  Research source}

Advertisement

Method 2

Method 2 of 2:

Using Gauss’s Technique

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/84\/Add-Consecutive-Integers-from-1-to-100-Step-5-Version-2.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-5-Version-2.jpg","bigUrl":"\/images\/thumb\/8\/84\/Add-Consecutive-Integers-from-1-to-100-Step-5-Version-2.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-5-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

1
Divide the series into two equal groups. To find out how many numbers are in each group, divide the number of numbers by 2. In this instance, since the series is 1 to 100, you would calculate $100\div 2=50$ . ^{[5]
X
Research source}
- So, the first group will have 50 numbers (1-50).
- The second group will also have 50 numbers (51-100).
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/c\/ce\/Add-Consecutive-Integers-from-1-to-100-Step-6-Version-2.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-6-Version-2.jpg","bigUrl":"\/images\/thumb\/c\/ce\/Add-Consecutive-Integers-from-1-to-100-Step-6-Version-2.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-6-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

Write the numbers in a row, beginning with 1 and ending with 50. ^{[6]
X
Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/96\/Add-Consecutive-Integers-from-1-to-100-Step-7.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-7.jpg","bigUrl":"\/images\/thumb\/9\/96\/Add-Consecutive-Integers-from-1-to-100-Step-7.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-7.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

Write these numbers in a row under the first group. Begin so that 100 lines up under 1, 99 lines up under 2, etc. ^{[7]
X
Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/4\/46\/Add-Consecutive-Integers-from-1-to-100-Step-8.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-8.jpg","bigUrl":"\/images\/thumb\/4\/46\/Add-Consecutive-Integers-from-1-to-100-Step-8.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-8.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

This means you will calculate $1+100=101$ , $2+99=101$ . etc. You don’t actually have to add up all the sets of numbers, because you should see that each set adds up to 101. ^{[8]
X
Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/0\/00\/Add-Consecutive-Integers-from-1-to-100-Step-9.jpg\/v4-460px-Add-Consecutive-Integers-from-1-to-100-Step-9.jpg","bigUrl":"\/images\/thumb\/0\/00\/Add-Consecutive-Integers-from-1-to-100-Step-9.jpg\/aid1275929-v4-728px-Add-Consecutive-Integers-from-1-to-100-Step-9.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

To find the sum of consecutive numbers 1 to 100, you multiply the number of sets (50) by the sum of each set (101): $101(50)=5050.$ So, the sum of consecutive number 1 through 100 is 5,050. ^{[9]
X
Research source}

Advertisement

Community Q&A

Search

Add New Question

Question

What is the total if I add up all numbers from 1 to 300?

Donagan

Top Answerer

See Sum the Integers from 1 to N .

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 5 Helpful 19
Question

What are the consecutive numbers that add up to 100?

Community Answer

You need to know how many consecutive numbers you are looking for and then use algebra to solve. For example, if you know you are looking for 5 numbers, you would set up the equation x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 100. Then, simplify: 5x + 10 = 100; 5x = 90; x = 18. So, the five consecutive numbers are 18, 19, 20, 21, and 22.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 7 Helpful 16
Question

The sum of four consecutive numbers is 50. What is the second number?

Community Answer

Let n be the first number, so that (n + 1) is the number you're looking for. The problem can be stated as (n) + (n+1) + (n+2) + (n+3) = 50. Combining terms, 4n + 6 = 50. Then 4n = 44, and n = 11. That means the number you're looking for is 12.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 0 Helpful 0

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Advertisement

Video

Tips

Submit a Tip

All tip submissions are carefully reviewed before being published

Thanks for submitting a tip for review!

You Might Also Like

Advertisement

References

About This Article

Reviewed by:

Math Teacher

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 107,319 times.

35 votes - 78%

Co-authors: 9

Updated: September 30, 2022

Views: 107,319

Categories: Algebra

In other languages

Thanks to all authors for creating a page that has been read 107,319 times.

Did this article help you?

Advertisement

X

-

-

1809

.mwimg a.image{display:none}#header_logo{display:none}#cse-search-box{display:none}#cse-search-box-noscript{display:block !important}#cse-search-box-noscript .cse_q{color:#FFF;background-color:#B3CE9C}.wh_search .cse_q{width:88%}#mw-mf-main-menu-button{display:none;visibility:hidden}.header .search{display:none}#search_footer .cse_sa{background-image:url(/extensions/wikihow/mobile/images/white_mag.png);background-size:22px}.related_box{display:block;background:#EEE;border:5px solid #FFF;vertical-align:bottom;width:127px;background-size:180px;float:left;height:120px;position:relative}.related_box p{line-height:15px !important;font-size:13px;background-color:rgba(67,95,44,.4);color:#FFF;font-weight:bold;padding:8px;margin:0 !important;position:absolute;bottom:0}#noscript_header_logo{display:table-cell;vertical-align:middle;background-position:50% 77%;background-repeat:no-repeat;background-image:url(/extensions/wikihow/mobile/images/wikihow_logo_230.png);background-size:105px;width:auto;height:51px;padding:0 14px}.articleinfo{display:none}#iorg_free_data{z-index:50;width:100%;position:fixed}#ara_recommendations{display:none}

Design a Mobile Website

View Site in Mobile | Classic

Share by: