How to Add 5 Consecutive Numbers Quickly

Co-authored by Christopher M. Osborne, PhD

Last Updated: February 28, 2023 Fact Checked

You might see “162 + 163 + 164 + 165 + 166 = ?” on a math quiz, or hear “Bet you can’t add up the five numbers from 162 through 166” from a friend who likes to boast about their math skills. But guess what — you can do it, and quickly! There are some simple math tricks that make adding up any 5 consecutive numbers easy. You can use them to wow your friends or impress your math teacher!

Method 1

Method 1 of 3:

Multiplying the Middle Number by 5

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Let’s say you’re given the numbers 51, 52, 53, 54, and 55. 53 is the middle number in this case, since it has two numbers before and after it in the series.^{[1]
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Multiply the middle number by 5 in your head. This is easier if you’re given small numbers like 1 through 5, of course (since 3 x 5 = 15). But it’s not that much more difficult to mentally multiply 53 x 5 = 265.^{[2]
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- First, separate 53 into 50 and 3.
- Second, multiply 50 x 5 = 250.
- Then, multiply 3 x 5 = 15.
- Finally, add those two results together: 250 + 15 = 265.
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The sum of any 5 consecutive whole numbers will always be divisible by 5, which means it will end in either 5 or 0. So, if your mental math gave you a result of 264 or 266, you know you need to recalculate.^{[3]
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Try it with larger numbers or negative numbers too. For instance, say your consecutive numbers run from 1,263 through 1,267. To mentally multiply the middle number, 1,265, you could add up 5,000 (= 1,000 x 5), 1,000 (= 200 x 5), 300 (= 60 x 5), and 25 (= 5 x 5) to get 6,325 as the sum of 1,263 through 1,267.^{[4]
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- Or, let’s say that your given series is -3, -2, -1, 0, 1. You still multiply the middle number by 5 to get your result — in this case, -1 x 5 = -5, which is also the sum of (-3) + (-2) + (-1) + 0 + 1.
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Grab a pencil and paper to see how it works. You can figure out the sum of any series of consecutive whole numbers by using the following formula, in which x = the sum, a = the first number in the series (e.g., 51), and n = the last number in the series (e.g, 55):^{[5]
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- x = (n x ((n + 1) / 2)) - ((a - 1) x (a / 2))
- x = (55 x (56 / 2)) - (50 x 25.5)
- x = (55 x 28) - 1275
- x = 1540 - 1275
- x = 265 (and 265 / 5 = 53, the middle number)

Method 2

Method 2 of 3:

Using the Highest or Lowest Number in the Series

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1
Pick any 5 consecutive numbers. For a simpler example (Example A), try 11, 12, 13, 14, 15.
- For more of a challenge (Example B), try 232, 233, 234, 235, 236.
For Example A, 15 x 5 = 75. For Example B, 236 x 5 = 1,180.
For Example A, 75 - 10 = 65. For Example B, 1,180 - 10 = 1,170.
For Example A, (11 x 5) + 10 = 55 + 10 = 65. For Example B, (232 x 5) + 10 = 1,160 + 10 = 1,170. As you can see, you end up with the same results.
In Example A, 13 is the middle number, and 13 x 5 = 65. In Example B, 234 is the middle number, and 234 x 5 = 1,170. To do Example B in your head, remember that you can break it up: (200 x 5 = 1,000) + (30 x 5 = 150) + (4 x 5 = 20) = 1,000 + 150 + 20 = 1,170.^{[6]
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Use the formula to further confirm your result. Use x = (n x ((n + 1) / 2)) - ((a - 1) x (a / 2)), where x = the sum, n = the highest number, and a = the lowest number.^{[7]
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- For Example A:
  - x = (15 x (16 / 2)) - (10 x 5.5)
  - x = (15 x 8) - 55
  - x = 120 - 55 = 65
- For Example B:
  - x = (236 x (237 / 2)) - (231 x 116)
  - x = (236 x 118.5) - 26796
  - x = 27966 - 26796 = 1170

Method 3

Method 3 of 3:

Adding Up Other Amounts of Consecutive Numbers

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1
Add four consecutive numbers by multiplying the highest by 4 and subtracting 6. For example, try 11, 12, 13, 14; 14 x 4 = 56, and 56 - 6 = 50.
- Or, you can multiply the sum of the middle two numbers by two. In the above case, 12 + 13 = 25, and 25 x 2 = 50.^{[8]
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Add six consecutive numbers by multiplying the highest by 6 and subtracting 15. Say you have a consecutive series from 11 through 16. You’d multiply 16 x 6 = 96, then subtract 15 to get 81.
- Alternatively, you can multiply the sum of the middle two numbers by 3. In the above case, 13 + 14 = 27, and 27 x 3 = 81.^{[9]
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Add seven consecutive numbers by multiplying the middle number by 7. If your series runs from 11 through 17, 14 is the middle number. Thus, 14 x 7 = 98.^{[10]
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- For any odd number (3, 5, 7, etc.) of consecutive numbers, you can always just multiply the middle number by the total number in the series.
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Add eight consecutive numbers by multiplying the highest by 8 and subtracting 28. If your numbers run from 11 through 18, multiply 18 x 8 = 144, then subtract 144 - 28 = 116.
- You can also multiply the sum of the middle two numbers by 4. In the above example, 14 + 15 = 29, and 29 x 4 = 116.^{[11]
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What are five consecutive whole numbers that add up to 50?

Donagan

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Let n be the first number. Then (n) + (n+1) + (n+2) + (n+3) + (n+4) = 50. Combining terms, 5n + 10 = 50. Then 5n = 40, and n = 8. The numbers are 8, 9, 10, 11 and 12.

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What are 5 consecutive whole numbers that equal 15?

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First divide 15 by 5 to get the middle number in the sequence. Since there are 5 numbers in the sequence used, we divide by 5. This gives us the number three. We know that this is the middle number in the sequence, so there are an equal amount of numbers both above and below this. Therefore, the numbers must be 1,2,3,4,5. Those five numbers add up to 15 to confirm that this answer is correct.

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The sum of five consecutive numbers is 100. How do I find the first number?

Donagan

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Let n be the first number. Then the next four numbers will be n+1, n+2, n+3, and n+4. Add the five numbers together and set them equal to 100: (n) + (n+1) + (n+2) + (n+3) +(n+4) = 100. Add together the five terms on the left to get 5n + 10 and let that equal 100: 5n +10 = 100. Subtract 10 from both sides: 5n = 90. Divide both sides by 5: n = 18. That's the number you're looking for.

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Tips

You can add any sequence of consecutive numbers, even or odd, regardless of the number of integers the sequence contains, by adding the first and the last number in the sequence, dividing it by two and multiplying the answer by the number of integers in the sequence. In algebra, we can say ((a + b) / 2) x n, or rearrange it to (n x (a + b)) / 2 to remove a set of parenthesis.
- For instance, say your series is 10, 12, 14, 16, and 18. Plug the numbers into the formula to get x = 5 x (10 + 18) / 2; x = 5 x 28 / 2; x = 5 x 14; x = 70.
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