How to Add Consecutive Integers from 1 to 100

Reviewed by Joseph Meyer

Last Updated: September 30, 2022 References

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]
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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

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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]
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2
Plug the values into the formula. This means substituting the first term in the series for $a_{1}$ $a_{{1}}$ , and the last term in the series for $a_{n}$ $a_{{n}}$ . When adding consecutive numbers 1 through 100, $a_{1}=1$ $a_{{1}}=1$ and $a_{n}=100$ $a_{{n}}=100$ .
- Thus, your formula will look like this: $S_{100}=n({\frac {1+100}{2}})$ .
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$Step 3 Add the values in the numerator of the fraction, then divide by 2.$

Since $100+1=101$ , you will divide 101 by 2: ${\frac {101}{2}}=50.5$ .
$Step 4 Multiply by n {\displaystyle n} .$

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, $n=100$ $n=100$ . So, you would calculate $100(50.5)=5050$ $100(50.5)=5050$ . Thus, the sum of the consecutive numbers between 1 and 100 is 5,050.^{[3]
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- To quickly multiply a number by 100, move the decimal point two places to the right.^{[4]
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Method 2

Method 2 of 2:

Using Gauss’s Technique

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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 $100\div 2=50$ $100\div 2=50$ .^{[5]
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- So, the first group will have 50 numbers (1-50).
- The second group will also have 50 numbers (51-100).
Write the numbers in a row, beginning with 1 and ending with 50.^{[6]
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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]
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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]
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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]
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What is the total if I add up all numbers from 1 to 300?

Donagan

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See Sum the Integers from 1 to N.

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What are the consecutive numbers that add up to 100?

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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.

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

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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.

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