How to Sum the Integers from 1 to N

Co-authored by Jessica Gibson

Last Updated: September 27, 2022 References

If you're preparing to take a standardized test or just want to sum numbers quickly, learn how to add the integers from 1 to $n$ . Since integers are whole numbers, you won't need to worry about fractions or decimals. Just decide which formula will help you answer your problem. Then plug the integer from the problem into the $n$ place and solve the equation.

Method 1

Method 1 of 2:

Assessing Your Sequence

1
Identify the arithmetic sequence. Look at the range of numbers you're trying to add together. If you'd like to use a formula to sum the integers, ensure that the numbers progress in a constant amount.^{[1]
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- For example, the series, 5, 6, 7, 8, 9 is a series and so is 17, 19, 21, 23, 25.
- You wouldn't be able to use 5, 6, 9, 11, 14 because the progression isn't constant.
2
Define $n$ for your sequence. In order to use a formula to find the sum of 1 to $n$ $n$ , choose the largest integer to be $n$ $n$ .
- For example, if you're trying to add all of the integers from 1 to 100, $n$ will be 100 because it's the largest integer in the sequence.
- As a reminder, integers are whole numbers, so $n$ cannot be a decimal, fraction, or negative number.
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3
Identify how many integers you're adding. In order to sum the integers from your starting number to $n$ $n$ , determine how many terms you're adding. For example, if you're adding the first 200 integers, you'll have 200 plus 1 to equal 201 integers.^{[2]
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- If you're adding the first integers from 1 to 12, you'll have 12 plus 1 to equal 13 terms.
4
Decide if you're adding exclusively. You may be asked to find the sum of a range of integers between two integers. If you're summing exclusively, you'll need to subtract 1 from your $n$ $n$ .^{[3]
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- For example, if you're finding the sum of the integers from 1 to 100 exclusively, subtract 1 from 100 to get 99.

Method 2

Method 2 of 2:

Using Formulas to Add the Integers

1
Define your formula for consecutive integers. Once you've defined $n$ $n$ as the largest integer you're adding, plug the number into the formula to sum consecutive integers: sum = $n$ $n$ ∗( $n$ $n$ +1)/2.^{[4]
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- For example, if you're summing the first 100 integers, plug 100 into $n$ to get 100∗(100+1)/2.
- If you're finding the first 20 integers, use 20 for $n$ . Work 20∗(20+1)/2 to get 420/2. Your answer will be 210.
2
Set up a formula to calculate only even integers. If the problems asks you to find the sum of only the even integers in a sequence starting with 1, you'll need to use a different formula. Plug your highest integer into $n$ $n$ so: sum = $n$ $n$ ∗( $n$ $n$ +2)/4.^{[5]
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- For example, if the problem asks you to find the sum of even integers from 1 to 20, use 20 as $n$ . Your formula will be 20∗22/4.
3
Define a formula to find the sum of odd integers. If the problems asks you to find the sum of only the odd integers, you'll need to find $n$ $n$ first. To find $n$ $n$ , add 1 to the highest number of the sequence. Then use it in this formula: sum = ( $n$ $n$ +1)∗( $n$ $n$ +1)/4.^{[6]
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- For example, to add the odd integers from 1 to 9, add 1 to 9. The equation will now look like 10∗(10)/4. Once you've worked the equation, you'll get 10∗(10)/4 to equal 25.
4
Work any of your defined formulas to find the sum. Once you've plugged in the integer, multiply the integer by itself plus 1, 2 , or 4 depending on your formula. Then divide your result by 2 or 4 to get the answer.^{[7]
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- For the example of consecutive formula 100∗101/2, multiply 100 by 101 to get 10100. Divide this by 2 to get an answer of 5050.
- For the example of even integers 20∗22/4, multiply 20 by 22 to get 440. Divide this by 4 to get a result of 110.

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What is the sum of the first n positive integers?

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The formula for the sum of the first n positive integers is n(n+1)/2. This formula works for consecutive integers.
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What is the sum of squares formula?

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The sum of squares is also known as the sum of squared deviation scores. The formula for calculating the sum of squares for a single set of scores is SS = ΣX^2 - ((ΣX)^2 / N).
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How do you find the sum of n terms?

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To find the sum of a series of arithmetic terms (that is, terms that increase or decrease by a constant amount each term), use the formula S˯n = n(a˯1 + a˯n)/2, where n is the number of terms, a˯1 is the first term in the sequence, and a˯n is the last term in the sequence.