In a math test you can find exercises like "162 + 163 + 164 + 165 + 166 =?". Perhaps a friend who likes to show off their number skills has once told you "I bet you can't add the five numbers 162 to 166". Guess what: you can do it, and fast! There are a few simple math tricks that allow you to easily add 5 consecutive random numbers. Use them to surprise your friends or impress your math teacher!
Steps
Method 1 of 3: Multiply the middle number by 5
Step 1. Identify the middle number in the series
Suppose you have the numbers 51, 52, 53, 54, and 55. In this case, the middle number is 53 since there are two numbers before it and two numbers after it.
Step 2. Mentally multiply the middle number by 5
Obviously it is much easier if you have to do it with small numbers like, for example, from 1 to 5 (3 x 5 = 15). But it is not that difficult to multiply 53 x 5 = 265 either.
- First, separate the 53 into 50 on one side and 3 on the other.
- Second, multiply 50 x 5 = 250.
- Then multiply 3 x 5 = 15.
- Finally, add the two results: 250 + 15 = 265.
Step 3. Confirm that the result ends in 0 or 5
The sum of any 5 consecutive numbers is always divisible by 5, which means that it must end in 0 or 5. Therefore, if the sum you made mentally results in 264 or 266, you will have to do it again.
Step 4. Try larger numbers or negative numbers
For example, think of the consecutive numbers from 1263 to 1267. To mentally multiply the middle number, 1265, you can add 5000 (= 1000 x 5), 1000 (= 200 x 5), 300 (= 60 x 5), and 25 (= 5 x 5) and you will get 6325 as a result of adding 1263 to 1267.
Your series could also be -3, -2, -1, 0, 1. In that case you will still have to multiply the middle number by 5 to arrive at the result. In this case, -1 x 5 = -5. If you add (-3) + (-2) + (-1) + 0 + 1, you will get the same result
Step 5. Take a pencil and paper to see how it works
You can find the sum of any series of consecutive whole numbers using the following formula, in which x is the sum, to is equal to the first number in the series (for example, 51) and is equal to the last number in the series (for example, 55).
- 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 of 3: Use the largest or smallest number in the series
Step 1. Pick 5 consecutive numbers at random
To start with a simple example (example A) imagine the numbers 11, 12, 13, 14 and 15.
For a more challenging example (example B), imagine the numbers 232, 233, 234, 235, and 236
Step 2. Multiply the largest number by 5
In the case of Example A, 15 x 5 = 75. In the case of Example B, 236 x 5 = 1180.
Step 3. Subtract 10 from the result to get the sum of the series
In example A, 75 - 10 = 65. In example B, 1180 - 10 = 1170.
Step 4. Use the smallest number by multiplying it by 5 and adding 10 to it
In the case of example A, (11 x 5) + 10 = 65. In the case of example B, (232 x 5) + 10 = 1160 + 10 = 1170. As you can see, the final result is the same.
Step 5. Compare the result by multiplying the middle number by 5
In example A, the middle number is 13 and 13 x 5 = 65. In example B, the middle number is 234 and 234 x 5 = 1170. If you want to solve example B mentally, remember that you can break it down: (200 x 5 = 1000) + (30 x 5 = 150) + (4 x 5 = 20) = 1000 + 150 + 20 = 1170.
Step 6. Use the formula to verify the result
Use x = (n x ((n + 1) / 2)) - ((a - 1) x (a / 2)), where x is the sum, to is the largest and smallest number.
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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
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For example B:
- x = (236 x (237/2)) - (231 x 116)
- x = (236 x 118, 5) - 26796
- x = 27966 - 26796 = 1170
Method 3 of 3: Add other amounts of consecutive numbers
Step 1. Add four consecutive numbers by multiplying the largest number by 4 and subtracting 6 from it
For example, in the series 11, 12, 13, 14; 14 x 4 = 56, and 56 - 6 = 50.
You can also multiply the sum of the two middle numbers by two. In the previous case, 12 + 13 = 25, and 25 x 2 = 50
Step 2. Add six consecutive numbers by multiplying the largest number by 6 and subtracting 15 from it
Suppose you have a series of consecutive numbers from 11 to 16. Multiply 16 x 6 = 96 and subtract 15. The result is 81.
Alternatively, you can multiply the sum of the two numbers in the middle. In the previous case, 13 + 14 = 27 and 27 x 3 = 81
Step 3. Add seven consecutive numbers by multiplying the middle number by 7
If your series goes from 11 to 17, the middle number is 14. Therefore, 14 x 7 = 98.
The sum of any series made up of an odd number of consecutive numbers (3, 5, 7, etc.) can always be obtained by multiplying the middle number by the number of numbers in the series
Step 4. Add eight consecutive numbers by multiplying the largest number by 8 and subtracting 28 from it
If you have the numbers 11 through 18, multiply 18 x 8 = 144 and then subtract 144 - 28 = 116.
You can also multiply the sum of the two middle numbers by 4. In the previous case, 14 + 15 = 29 and 29 x 4 = 116
Advice
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The sum of any sequence of consecutive numbers (either even or odd) can be calculated, regardless of the number of integers it contains, by adding the first and last numbers in the sequence, dividing the result by 2 and multiplying that value by the number of integers in the sequence. In algebraic terms, this could be expressed as: ((a + b) / 2) x n. This can be regrouped in (n x (a + b)) / 2 to remove a pair of parentheses.
For example, if your series is 10, 12, 14, 16 and 18, replace the numbers in the formula and you will get the result of the sum: x = 5 x (10 + 18) / 2; x = 5 x 28/2; x = 5 x 14; x = 70