Series Completion Problems: Definition & Strategies

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Read this lesson to learn how you can solve series completion problems. Learn how to spot a pattern which you can continue as well as how to find if something is wrong with a given series.


A series is a string of items. In math, a series usually involves a string or sequence of numbers. Some series have a pattern and others are random with no pattern. For example, the counting numbers is a series with a pattern, each term has 1 added to the previous number.

  • 1, 2, 3, 4, 5, …

Most series with a pattern will keep going indefinitely. Like the counting numbers, you can keep counting forever, you'll never finish counting as there will always be one more.

Your counting numbers is a series
series completion problems

In math, you'll come across series problems that test your logic skills. Some will ask you to find a pattern and then to complete the series either by finding a missing term or continuing the given series. Other problems will ask you to find which term is the wrong term. To solve these kinds of problems, your first and biggest task is that of finding a pattern. To find a pattern, look for multiplication by a specific number each time or addition of a specific number each time. Sometimes, you may also see division or subtraction with a specific number. Either way, you will have to look for a mathematical operation that is repeated.

Complete the Series

First, let's take a look at problems that ask you to complete a given series.

Complete this series.

  • 2, 4, 6, 12, 14, 28, ?

This problem has a question mark at the end, so it wants you to find the term that comes after the 28. First, you need to find a pattern. Look at the numbers and see if you can see a relationship between the terms. Then, examine if this relationship is repeated. Looking closely at these numbers, you might notice that the second term ''4'' is two times the first term. But the third term 6, is 2 more than the second term 4. You keep looking and you now see a pattern. The pattern goes like this: times 2, then plus 2, then times 2, then plus 2 again. So, the term that goes after the sixth term 28, is ''30'' since you already had a times 2 to get from the 14 to the 28. Now, you need to add 2: 28 + 2 = 30. And you are done.

This example shows that not all series have the same relationship term after term. Sometimes, you'll have two different relationships that appear at every other term as in this example that had the relationships of times 2 and plus 2. And these relationships aren't always addition or multiplication. They could be division or subtraction as well.

What's Wrong?

Now, let's look at solving problems that ask you to find the wrong term.

Which term does NOT belong?

  • 3, 7, 9, 12, 15

To solve this type of problem, you likewise have to first find a pattern. Once you've found your pattern, you can then see which term does NOT belong.

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