Historical Context & Motivation
People have been fascinated by patterns for thousands of years. Ancient builders used repeating designs on walls. Early farmers noticed patterns in the seasons. Mathematicians noticed patterns in numbers, too, and began writing rules to describe them.
Understanding patterns lets you predict what comes next without counting every single item. That skill saves time and helps you solve real problems, from planning a budget to cracking a test question.
The big question these thinkers were all asking is the same question you will explore in this lesson: When you see a list of numbers, how do you figure out the rule and predict what comes next?
Core Principles & Definitions
Before you start solving pattern problems, you need to know a few key ideas. These are the building blocks for everything else in this lesson.
Sequence
Common Difference
Common Ratio
Pattern Rule (Expression)
Visual Explanation — Seeing the Pattern
The best way to understand a pattern is to see it. The diagram below shows an arithmetic sequence — the kind where you add the same number each time.
Notice that the difference between every pair of neighbors is exactly 4. That steady jump is what makes this an arithmetic sequence. If the differences changed each time, you would need a different type of rule.
Mathematical Framework
Once you spot the pattern type, you can write a formula (a math rule) to find any term without listing them all. Here are the two main formulas you need.
Use this formula when the sequence grows (or shrinks) by adding the same amount each time. For example, 3, 7, 11, 15, … has a(1) = 3 and d = 4.
Use this formula when the sequence grows (or shrinks) by multiplying by the same amount each time. For example, 2, 6, 18, 54, … has a(1) = 2 and r = 3.
Types of Patterns You'll See on the TACHS
TACHS pattern questions come in several flavors. Knowing the different types helps you pick the right strategy quickly. The diagram below compares the two most common types side by side.
| Pattern Type | How It Works | Example |
|---|---|---|
| Arithmetic | Add (or subtract) the same number each time | 4, 9, 14, 19, 24, … (d = +5) |
| Geometric | Multiply (or divide) by the same number each time | 3, 12, 48, 192, … (r = ×4) |
| Repeating | A group of numbers cycles over and over | 1, 4, 7, 1, 4, 7, 1, 4, 7, … |
| Other / Two-Step | Uses a combination of operations (add then multiply, etc.) | 2, 5, 11, 23, … (×2 + 1) |
Worked Example
Let's walk through a complete example, just like you'd see on the TACHS. Pay attention to each step so you can repeat the process on your own.
Strategies, Strengths & Pitfalls
Different strategies work better for different pattern types. Knowing when to use each one will help you work faster and avoid common mistakes.
| Strategy | When to Use It | Watch Out For |
|---|---|---|
| Subtract consecutive terms | When you think the pattern adds the same number each time (arithmetic) | Don't assume it's arithmetic until you check at least 2 or 3 differences |
| Divide consecutive terms | When differences are not equal — the pattern might multiply (geometric) | If the ratio doesn't stay the same, try a two-step rule or look for a repeating cycle |
| Use the formula | When you need a term far down the line (like the 50th term) | Remember to use (n − 1), not just n, in the formula |
| Make a table | When the pattern seems tricky or uses more than one operation | Double-check each row before moving on |
Connection to Advanced Topics
The pattern skills you are learning now are the same ones used in high school algebra and beyond. When you study linear functions in algebra, you will see that an arithmetic sequence is really just a straight line in disguise. And geometric sequences lead to exponential functions, which model things like population growth and compound interest.
| What You Learn Now | Where It Goes Next |
|---|---|
| Arithmetic sequences (add a constant) | Linear functions: y = mx + b, slope, graphing lines |
| Geometric sequences (multiply by a constant) | Exponential functions: y = a × rˣ, compound interest, growth/decay |
| Writing pattern rules with n | Writing and evaluating algebraic functions, input-output tables |
| Finding the common difference d | Slope of a line (rise over run), rate of change |
So by mastering patterns now, you're actually getting a head start on some of the biggest topics in algebra. The common difference d will become the slope of a line, and the first term a(1) will connect to the y-intercept. Pretty cool, right?
Practice Problems
Try these five problems on your own. They start easy and get harder. After each one, check the answer to see if you're on the right track.
Lesson Summary
A pattern is a sequence of numbers that follows a rule. In an arithmetic sequence, you add (or subtract) the same common difference each time. In a geometric sequence, you multiply (or divide) by the same common ratio each time. To find the type, subtract or divide consecutive terms and check if the result stays constant.
Use the formula a(n) = a(1) + (n − 1) × d for arithmetic sequences and a(n) = a(1) × r⁽ⁿ⁻¹⁾ for geometric sequences to jump straight to any term. Some patterns use two-step rules or repeating cycles. Always check your answer by listing a few terms to make sure the rule works. These skills connect directly to linear and exponential functions in high school algebra.