What Is the Sum of 1 to 100

November 24, 2021

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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] 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.

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Write the formula for finding the sum of an arithmetic series. 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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Add the values in the numerator of the fraction, then divide by 2. Since $100+1=101$ , you will divide 101 by 2: ${\frac {101}{2}}=50.5$ .
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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 $100\div 2=50$ .^[4]
- So, the first group will have 50 numbers (1-50).
- The second group will also have 50 numbers (51-100).
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Write the first group, 1-50, in ascending order. Write the numbers in a row, beginning with 1 and ending with 50.
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Write the second group, 100-51, in descending order. Write these numbers in a row under the first group. Begin so that 100 lines up under 1, 99 lines up under 2, etc.
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Add each vertical set of numbers. 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.^[5]
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Multiply 101 times 50. 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.

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Question

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

What are the consecutive numbers that add up to 100?

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.