How to solve number series

Number series are used in a variety of logic tests. You are given a sequence of numbers, and your task is to append the number that should logically follow. In this article, we will demonstrate how to solve such number series.

Assignment

The classical problem for number sequences looks like this:

$$2, 4, 6, 8, 10, ?$$

Your task is to find the number that can be added after the question mark in a way that makes the most sense. Thus, you must identify a characteristic property among the given numbers and use it to calculate the missing number. In this case, the result will be the number 12, because each number in the series is always 2 greater than the previous one.

The solution may not always be straightforward. We could say that any number can follow the question mark, as long as we can justify this choice.

Addition and multiplication

Among the simplest series are those where you simply add a number. Examples include:

$$3, 10, 17, 24, ?$$

Here, 7 is always added, so in place of the question mark, it would be 31. Here is another example:

$$-8, -5, -2, 1, ?$$

We always add 3; the next number in the series will be 4. Another example:

$$17, 8, -1, -10, ?$$

In this row, we add −9, or subtract 9. The next number in line will be −19. Next example:

$$2, 4, 8, 16, ?$$

Here we are no longer adding; we are multiplying. The number is always twice as large as the previous one, so the result will be 32. Next example:

$$27, 9, 3, 1, ?$$

In this series, by contrast, we are dividing by three. The result will be $\frac13$. We can also add a non-constant number:

$$2, 4, 7, 9, 12, ?$$

We alternate adding +2 and +3. So we add +2 to 2 to get 4. Then we add +3 to get 7, and so on. Adding +2 to 12 results in 14. Another example:

$$5, 6, 8, 11, 15, 20, ?$$

In this series, we always add one more number than in the previous step, starting from +1. Thus 5 + 1 = 6, 6 + 2 = 8, 8 + 3 = 11, etc. Finally, we add 6, resulting in 20 + 6 = 26. Here's another example:

$$1, 2, 6, 12, 36, 72, ?$$

We alternate between multiplying by two and three: 1 · 2 = 2, 2 · 3 = 6, 6 · 2 = 12, and so on. Ultimately, we multiply the final value by three, resulting in 216.

Alternating rows

We can further combine rows and operations, using addition once and multiplication a second time:

$$3, 12, 14, 56, 58, 174, ?$$

First, we multiply by four and then add two in the subsequent step. Thus: 3 · 4 = 12, 12 + 2 = 14, 14 · 4 = 56, etc. Finally, we add two, so the result is 176. A similar example:

$$3, 9, 5, 15, 11, 33, 29, ?$$

First, multiply by three, then subtract four. Thus: 3 · 3 = 9, 9 − 4 = 5, 5 · 3 = 15, etc. Finally, multiply by three to obtain the result of 87.

We can combine the rows such that there are actually two rows, each with its own distinctive property. For example:

$$2, 1, 4, 6, 8, 11, 16, 16, 32, ?$$

In the odd positions, we have the familiar series 2, 4, 8, 16, 32, where each number is always twice the previous one, and in the even positions, we have the sequence 1, 6, 11, 16, ?. This is a series where we simply add five each time. Therefore, the result is 16 + 5 = 21. Here is another example:

$$2, 7, 3, 11, 5, 15, 8, 19, 12, ?$$

In this series, we can identify two sub-series. At odd positions, we have 2, 3, 5, 8, 12, which forms a sequence where each number is one more than the sum added in the previous step. At even positions, we have 7, 11, 15, 19, ?; here we simply add four each time. The result is 23.

Classical series

There are several classical series that are worth mentioning. To begin with:

• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, … - This sequence represents the squares of natural numbers, which can also be written as 1², 2², 3², 4², 5², …. If you encounter any of these numbers in the sequence, particularly the larger ones, it is likely that the square root is playing a significant role.

• 0, 1, 1, 2, 3, 5, 8, 13, 21, … - This is the famous Fibonacci sequence. Each number is defined as the sum of the two preceding ones. Reversing the order: 21 = 8 + 13, 13 = 5 + 8, 8 = 3 + 5, etc. The first two numbers, zero and one, are fixed.

• 2, 3, 5, 7, 11, 13, 17, 19, … - this is a sequence of prime numbers. It is important to note that prime numbers do not include the number one.

• 1, 2, 6, 24, 120, 720, 5040, 40320, … - This is a sequence of factorials. The number at the n-th position is derived from the product 1 · 2 · 3 · … · n. For example, the number at the fourth position is 24, which is calculated as 1 · 2 · 3 · 4 = 24.

Other types of series

Example:

$$17, 3, 6, 14, -3, 23, 19, ?$$

The solution is that if we add the numbers in the first and second positions, we get the number twenty: 17 + 3 = 20. The same applies to the following pairs: 6 + 14 = 20, −3 + 23 = 20, 19+? = 20. Thus, in place of the question mark, there should be a one. Another example:

$$?, 7, 12, 4, 5, 9, 6, -1, 5$$

In this sequence, we split the numbers into triples a, b, c, with a + b = c. Thus, ?+7 = 12 and 4 + 5 = 9. Therefore, the number that replaces the question mark is 5. Here’s another example:

$$2, 4, 10, 28, ?$$

Here we have combined two operations: multiplication and addition. The next number is obtained by multiplying the previous one by three and subtracting two. So we have: 4 = 2 · 3 − 2, 10 = 4 · 3 − 2, 28 = 10 · 3 − 2. Next in line is 28 · 3 − 2 = 82.