A sequence is a list of numbers written in a special order such as (1, 2, 3, 4…), which typically follows a pattern. Sequences are usually set in brackets ( ) to notate the sequence, and each element (also known as a “member” or “term”) of the sequence is separated by a comma, like this:
(4, 5, 6, 7)
Finite and Infinite Sequences
A sequence can be finite or infinite, depending on whether or not it has a set end point.
If a sequence has a set beginning and end, it is a finite sequence:
(10, 11, 12, 13)
This finite sequence started at 10 and stopped at 13.
If a sequence continues to increase or decrease indefinitely, it is considered is an infinite sequence. An infinite sequences uses an ellipsis (…) to indicate that the sequence continues past the final number:
(10, 15, 20, 25, 30, 35…)
This infinite sequence will continue to increase by 5 forever.
Finding the Pattern
Once you recognize that you’re dealing with a sequence, you next need to determine what its pattern is. Sometimes, that’s quite simple:
(1, 2, 3, 4, 5, 6, 7…)
In this example, each new number is created by adding 1 to the previous number. The next number in this sequence is 8.
This is a very simple example of an arithmetic sequence. Arithmetic sequences involve adding or subtracting to achieve each new number. The following example is the opposite of the one above, in which you subtract 1 each time:
(5, 4, 3, 2, 1, 0, -1, -2…)
Arithmetic sequences can also be more complex. In some cases, they increase by a certain number:
(20, 40, 60, 80, 100…)
In this example, each new number is achieved by adding 20 to the previous number. This one began at 20, making it quite simple to determine the next number (it’s the next multiple of 20). But number sequences can begin at any number:
(3, 23, 43, 63, 83, 103…)
This is the exact same pattern, only with a different starting term.
So far we’ve discussed sequences where each consecutive term is obtained by adding a set number to the previous term. But sequences can include a variety of operations. Consider the following sequence:
(1, 4, 16, 48…)
For each new term, you must multiply the last term by 4. 1×4 = 4, 4×4 = 16, etc.
This is called a geometric sequence, because you are multiplying by the same value each time.
You can also multiply by a value less than one:
(20, 10, 5, 2.5, 1.25…)
In this example, the common variable is ½. It’s also the same as dividing by 2.
Sequences don’t have to be tied to a single variable. You can create any number of variables, so long as they create a repeatable pattern. Consider this:
(1, 3, 2, 4, 3, 5, 4, 6…)
This sequence repeats a pattern of (+2, -1): 1+2= 3, 3-1= 2, 2+2= 4, etc.
Sequences can also be a mix of arithmetic and geometric:
(2, 6, 4, 12, 10, 30, 28…)
Can you identify the pattern? It’s a tricky one, because it combines multiplication and subtraction: (×3, -2): 2×3= 6, 6-2= 4, 4×3= 12, 12-2= 10, etc.
Number patterns are not tied to any specific rules. You can add, subtract, multiply, take the square root, cube a number, you name it! You can even do more than one operation for each term:
(1, 4, 10, 22, 46, 94)
In this example, each new term is created by multiplying the previous number by 2 and adding 2! Number patterns be as simple or as complex as your imagination can make them.
The Fibonacci Sequence
One of the most famous number patterns, the Fibonacci Sequence is actually one of the simplest to reproduce. Each new number is the sum of the two previous numbers in the sequence:
(1, 1, 2, 3, 5, 8, 13, 21…)
Since there will always be two previous numbers to add together, the sequence can go on forever.
There are plenty of resources available for students of any age who want to learn more about number sequences and/or test their ability to identify number patterns.
- Math is Fun: Common Number Patterns: this site presents several types of number patterns in an easily-accessible way. If you’re interested in exploring this subject further, this is a great place to start.
- Arithmetic Sequences and Series: this site is geared toward a slightly older audience. It goes much more in-depth into analyzing sequences and developing a formula for each.
- Spooky Sequences: this interactive game helps kids practice analyzing sequences and determining which number comes next.
- Study Jams Number Patterns: this site offers more advanced pattern recognition tests, along with explanations for how to determine each pattern. It’s light on instruction, but a great way to test your pattern identification skills.
If you’re looking for a more in-depth study of number patterns, there are plenty of books available for students, teachers, and general number enthusiasts.
- 300+ Mathematical Pattern Puzzles: Number Pattern Recognition & Reasoning (2015) by Chris McMullen: this collection of pattern puzzles will challenge and teach students of any age. Each chapter introduces a variety of new mathematical concepts, and then shows them in use each through a series of pattern examples.
- Patterns in Mathematics, Grades 3-6: Investigating Patterns in Number Relationships (2013) by Paul Swan: geared toward younger students, this book provides an introduction to mathematical patterns, both in terms of numbers and shapes.
- The Fabulous Fibonacci Numbers (2007) by Posamentier and Lehmann: this highly accessible text covers the long history of the Fibonacci sequences and the many ways the pattern occurs throughout the world, in art, nature, and even our financial markets.
Number patterns aren’t just fun to figure out; they are also a great way to learn to think mathematically. They force us to analyze sequences and apply different equations until we find the one that works. For young math students, they can be a great tool for learning addition and multiplication. For advanced students, sequences challenge them to think beyond the simple math problem. And for the rest of us, they can provide endless challenges, and plenty of fun.