Identifying Number Patterns in Math

When you look at a list of numbers or shapes, identifying the pattern means asking one question over and over: what changes from one term to the next, and does that change stay the same? If the change is consistent, you have found the rule. For example, in the numbers with patterns \(2, 4, 6, 8, 10\), each term is \(2\) more than the one before it. That constant amount of change is called the common difference, and you can check it with the formula \( d = a_n - a_{n-1} \), where \(a_n\) is a term and \(a_{n-1}\) is the term right before it.

Not every pattern grows by the same amount each time, and some patterns do not use numbers at all. It helps to sort what you see into a few categories.

• Repeating patterns cycle through the same group of items, like red, blue, blue, red, blue, blue.
• Growing patterns increase (or decrease) by a rule each time, such as \(3, 6, 9, 12\), where every term is \(3\) more than the last. If you want to practice building these from scratch, see creating growing patterns.
• Shape or picture patterns use figures instead of numbers, but the same idea applies: look for what stays the same and what changes.

Once a pattern is identified, you can go on to describe it in a table, a picture, or an equation. Rewriting a pattern in a different but equivalent form is covered separately in translating patterns.

Use the same short checklist any time you are asked to identify a pattern for math:

1. List the terms in order, without skipping any.
2. Find the difference (or ratio) between consecutive terms.
3. Check that this change is the same between every pair of terms.
4. Write the rule in your own words, such as "each term is the previous term plus four."
5. Test the rule on at least two terms you have not used yet to confirm it works.

Identify the pattern in \(5, 9, 13, 17, 21\).

Step 1: Subtract each term from the one after it: \(9 - 5 = 4\), \(13 - 9 = 4\), \(17 - 13 = 4\), \(21 - 17 = 4\). Step 2: Since every difference equals \(4\), this is a growing pattern with a common difference of \(4\). Step 3: The rule can be written as \( a_n = a_1 + (n-1)d \), which here becomes \( a_n = 5 + (n-1)(4) \). This same numeric pattern shows up constantly on tests that ask you to work with number patterns, so it is worth practicing until the steps feel automatic.

Sometimes a pattern is shown as a table connecting a position number to a term value.

Position (n)	1	2	3	4
Term value	3	6	9	12

Comparing the position number to the term value, each term is exactly \(3\) times its position: \(1 \times 3 = 3\), \(2 \times 3 = 6\), \(3 \times 3 = 9\), \(4 \times 3 = 12\). So the rule is "multiply the position number by 3." Tables like this are a quick way to organize numbers with patterns before writing a rule, and they set you up well for later work in extending patterns, where the same rule is used to predict terms far beyond the ones you can see.

Being able to identify patterns and math relationships quickly saves time on every problem that follows, because a confirmed rule tells you exactly what to expect next. Whether the question shows shapes, a table, or a plain list of numbers, the same habit of comparing consecutive terms and testing a rule will get you to the correct answer.

Related lessons

• Math 1