How to Sum the Positive Integers from 1 to n

X

This article is co-authored by a team of editors and trained researchers who confirm the accuracy and completeness of the article.

The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.

This article has been viewed 32,433 times.

Whether you’re preparing for an exam or just want to add lots of numbers quickly, you’ll be able to do it if you know how to add positive integers from 1 to 1. $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ . Since this is a set of natural numbers, you don’t need to care about fractions or decimals. Just choose the correct formula to do the calculation, then replace the integer in the problem $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ and solve the equation.

Table of Contents

Rating of the additive level

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/f/f4/Sum-the-Integers-from-1-to-N-Step-1-Version-4.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-1-Version-4.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/f/f4/ Sum-the-Integers-from-1-to-N-Step-1-Version-4.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-1-Version-4. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Additive recognition. Look at the range of numbers that you want to sum. If you want to use the formula to add multiple integers, the difference between the terms must be constant. ^{[1] X Research Source}

For example, the sequence of numbers 5, 6, 7, 8, 9 or 17, 19, 21, 23, 25 are additive levels.
We cannot apply the formula to 5, 6, 9, 11, 14 because this sequence of numbers is irregular.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/c/ce/Sum-the-Integers-from-1-to-N-Step-2-Version-4.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-2-Version-4.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/c/ce/ Sum-the-Integers-from-1-to-N-Step-2-Version-4.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-2-Version-4. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Find $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ of the additive level. To use the formula to sum integers from 1 to

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

, let’s choose the largest integer

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

.

For example, to add all integers from 1 to 100, $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ will be 100 because this is the largest integer in the set.
Again, we’re dealing with the set of positive integers, so $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ cannot be a decimal, fraction or negative number.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/c/ca/Sum-the-Integers-from-1-to-N-Step-3-Version-4.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-3-Version-4.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/c/ca/ Sum-the-Integers-from-1-to-N-Step-3-Version-4.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-3-Version-4. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Determines how many integers are in the arithmetic progression. To add integers from the starting number to

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

, determine how many terms to add. For example, if you are adding the first 200 positive integers, we get 200 + 1 = 201 terms. ^{[2] X Research Source}

For the sequence of positive integers from 1 to 12, we have 12 + 1 = 13 terms.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/6/6d/Sum-the-Integers-from-1-to-N-Step-4-Version-4.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-4-Version-4.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/6/6d/ Sum-the-Integers-from-1-to-N-Step-4-Version-4.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-4-Version-4. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Read the article carefully to see if you need to exclude or not. Maybe the problem will ask you to sum the terms in the range between two positive integers. To exclude, you need to get

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

– 1. ^{[3] X Research Source}

For example, to calculate the number of positive integers that range from 1 to 100, you take 100 – 1 = 99.

Apply the formula to add positive integers

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/2/2a/Sum-the-Integers-from-1-to-N-Step-5-Version-3.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-5-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/2/2a/ Sum-the-Integers-from-1-to-N-Step-5-Version-3.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-5-Version-3. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Determine the formula for a sequence of consecutive positive integers. After determining

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

(the largest integer in the series), substitute this value in the arithmetic sum formula: sum =

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

(

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

+1)/2. ^{[4] X Research Sources}

For example, sum the first 100 positive integers. Replace $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ = 100, substituting into the formula we get 100∗(100+1)/2.
If you are looking for the sum of the first 20 positive integers, replace $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ = 20. We have: 20∗(20+1)/2 = 420/2. So the sum of the first 20 positive integers is 210.

READ More: How to Admit to Foot Cramps

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/c/c5/Sum-the-Integers-from-1-to-N-Step-6-Version-2.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-6-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/c/c5/ Sum-the-Integers-from-1-to-N-Step-6-Version-2.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-6-Version-2. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Formula for sum of even positive integers. If the problem asks you to calculate only the sum of even positive integers in a sequence starting at 1, you need to use a different formula. Substitute the largest integer in

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

as follows: sum =

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

(

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

+2)/4. ^{[5] X Research Sources}

For example, calculate the sum of even integers from 1 to 20. When replacing $n$ ${displaystyle n}$ $How to Sum the Positive Integers from 1 to n$ = 20 into the formula, we have: 20∗22/4.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/9/9b/Sum-the-Integers-from-1-to-N-Step-7-Version-3.jpg/v4- 728px-Sum-the-Integers-from-1-to-N-Step-7-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/9/9b/ Sum-the-Integers-from-1-to-N-Step-7-Version-3.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-7-Version-3. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Determine the formula for the sum of odd integers. If the problem asks you to sum only odd numbers, first find

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

. To determine the value of

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

, you take the largest term of the sequence plus 1. Then, substitute this value into the formula: sum = (

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

+1)∗(

n

{displaystyle n}

How to Sum the Positive Integers from 1 to n

+1)/4. ^{[6] X Research Sources}

For example, let’s sum the odd integers from 1 to 9. First, we have n = 9 + 1 = 10. The equation will now be 10∗(10)/4 = 25.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/9/90/Sum-the-Integers-from-1-to-N-Step-8.jpg/v4-728px-Sum- the-Integers-from-1-to-N-Step-8.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/9/90/Sum-the-Integers-from- 1-to-N-Step-8.jpg/v4-728px-Sum-the-Integers-from-1-to-N-Step-8.jpg”,”smallWidth”:460,”smallHeight”:345,” bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Perform the calculation according to the formula you defined to find the sum. After substituting the integer in the formula, find the product of the integer by itself plus 1, 2, or 4 depending on the formula. Then, multiply the product by 2 or 4 to get the final answer. ^{[7] X Research Sources}

In the example that requires the sum of a sequence of consecutive numbers, we perform the calculation 100∗101/2 by taking 100 * 101 = 10100. Continue to divide this product by 2, the final result will be 5050.
In the example asking to calculate the sum of even integers, we have 20∗22/4, take 20 * 20 = 440. Divide this result by 4, the answer is 110.

Steps

Rating of the additive level

Apply the formula to add positive integers

Related Posts