Number sequences

Numerical tests are almost always part of your assessment, even if the job you are applying for has little to do with numbers. This is because potential future employers want to gain insight into all domains of your intelligence level (abstract, verbal, and numerical).

Number sequences are about recognizing numerical relationships. Doing simple calculations in your head is essential for this. Addition, subtraction, division, and multiplication are all involved, but it is a shame to waste a lot of time calculating, because you want to use that to determine the numerical relations.

In sequences like these, you can see a trend that gives you information about the order of magnitude of the answer. For example, look at the number series below.


In this simple sequence, 2 is added to the previous number. In this case, the answer is 12. If you graph this sequence, it creates a rising line and, because the same amount is added at each interval, the line is straight. If you know that there is a “monotonic relationship”, you can exclude certain answers because they may be lower than the last-known number of the sequence.

Of course, this trend may also move in a different direction as in the sequences below.


In this sequence, you get the next number by subtracting 10 from the previous number. In this case, the answer is 84 (94 minus 10). The monotonic relationship is present here, but now it creates a decreasing line.

Monotonic sequences always involve adding and subtracting as the operations. This makes them relatively easy, unless you are particularly bad at doing math in your head.

There are, of course, also complicated connections incorporated into the number sequences, where the increase or decrease is not uniform, but exponential instead. This means that numbers can quickly become larger or smaller. Such sequences are often a bit more difficult because the mathematical operations are harder to determine. This means that you will not notice them as quickly.

View the sequences below to understand an exponential relationship.


In this sequence, the previous number is multiplied by 3. Typically, numbers are not so large in number sequences, but you can clearly see in the graph that there is an exponential increase. The correct answer is 486. Compared to monotonic relationships, the increase or decrease is much larger and therefore somewhat more difficult to figure out. If there is a decreasing exponential trend, the operation is division.

To make things even more complicated, test makers can also use sequences where the connections alternate. For instance, they might mix a monotonic and exponential relationship. This results in differing operations. The series below is an example of this.


In this case, there is alternation between multiplication and addition. One number is multiplied by 7, while 1 is added to the next. These two operations are repeated. In this case, the correct answer is 400, specifically 399 plus 1. With this sequence, you can already see that knowing your multiplication tables is useful, because you need to know that 7 times 8 is 56 without thinking.

The arches method
The arches method is a great way to write out operations. This helps you make fewer mistakes and forces you to adhere to the pattern. When you arrive at an answer with this method, there is little doubt about whether it is correct. You simply know that it is right. This is different from other intelligence problems.

The sequence below is solved using the arches method. You can see that the sequence climbs, but starts with a zero. This lets you know that there is no multiplication in the first step, but what exactly is happening?

Third number is sum of the two previous numbers

0+3=3  3+3=6  3+9=15  9+15=24

You can start by simply writing out what you see. For example, you can see that a three is added repeatedly. However, the step from 9 to 15 proves this wrong. In short, you know that this is not the answer. 3 times 2 is 6, but this operation does not get you anywhere. For some sequences, the operation involves more than one number. That is also what is happening here. 0 plus 3 is 3. 3 plus 3 is 6. As it turns out, the third number is the sum of the two previous numbers. Knowing this, the correct answer is 24, because 9 plus 15 is 24.

With the arches method, you can solve sequences that you would never figure out if you kept staring at a sequence without writing anything down. You should work on viewing a series of numbers as a puzzle, where you work out different things with trial and error by using the arches method. The more often you do this, the more quickly you will solve these problems.

In the sequence below, it will take a while before you see what the pattern is at work. In any case, you can see that the sequence is decreasing. However, this is happening in separate steps. First, write down what is being subtracted during each step. 34 to 21 means that 13 is subtracted. 21 to 13 means that 8 is subtracted.

If you have a keen sense for these things, you will see that 13 and 8 also appear in the series. Be on the lookout for hints like these, because they will immediately open your eyes to the relationship, such as in this sequence.

In more complex sequences, there are often two independent sequences operating side by side. It is important that you be aware of this, because two independent sequences often occur in seemingly “inexplicable” problems. The sequence below has two independent sequences.

In the blue series, 2 is being added. That is a simple operation, but if you do not notice that there are two sequences you will not notice this all that quickly. The black sequence is always multiplied by 3. The correct answer here is 3 times 27 and that is 81. Of course, you can keep going through the sequence yourself and the number after 81 will clearly be 11 plus 2.

Practice, practice, practice
Sites where you can practice multiplication tables as well as other simple calculations are good for support. The less time you spend on calculations, the more time you have to deduce the relationships in the sequences. Furthermore, practice gives you more self-confidence, because the numbers no longer seem threatening.

In addition to improving the math you can do in your head, it also makes a lot of sense to create number sequences yourself. This allows you to check whether you have the correct sequence, because you can continue it yourself.

Tips for solving number sequences

  • Keep working! If you spend too much time on one task, you will lose time, which could be used on a problem that you can actually solve.
  • Use the arches method to recognize patterns.
  • Start by making notes on the sequence that you cannot figure out right away and describe what you are doing in your head. By making the steps in the process more explicit, you force yourself into taking a constructive approach instead of staring at the problem.
  • When using the arches method, write down the mathematical operation and not just the numbers, because then you will forget how a sequence begins.
  • Try to figure out the relationship in the sequence (rising/falling, monotonic/exponential/alternating).
  • Pay attention when you see big jumps and then small changes that alternate. Could there be two different sequences?
  • Starting at the end of the sequence can be helpful, because the size of the numbers makes it clear which operation is being applied.
  • Cross out the answers that do not make sense. These are multiple-choice questions, so you can exclude certain answers if you already know the relationship. This increases your chance of making a correct guess.

Practice makes perfect!

It is very important to practice for a capacity test. If you do not practice, your score may be lower, which often decreases your chances of getting that much-desired job! By practicing, you can solve problems more quickly and efficiently, so that your score will increase.

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