Historical Context & Motivation
People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain numbers follow a predictable order, and they used those patterns to solve real-world problems like measuring land and counting harvests. The idea of finding a pattern rule (a short description that tells you how to get from one number to the next) is one of the oldest ideas in math.
On the SHSAT, you will see sequences of numbers and be asked to figure out the rule behind them. If you can spot the pattern, you can predict any term in the sequence — even the 100th one! The big question is: How do you find the hidden rule inside a list of numbers?
Core Principles & Definitions
Before we dive into examples, let's make sure we understand the key vocabulary. A sequence is an ordered list of numbers that follow some rule. Each number in the list is called a term. The pattern rule is the instruction that tells you how terms are connected.
Sequence
Term
Common Difference
Common Ratio
Position Number (n)
Visual Explanation
The diagram below shows two common types of sequences. On the left is an arithmetic sequence where you add the same number each time. On the right is a geometric sequence where you multiply by the same number each time. Notice how the arrows show the operation connecting each term.
The most important thing the diagram shows is the constant change between terms. In the arithmetic sequence on the left, every arrow says "+ 4." In the geometric sequence on the right, every arrow says "× 3." When the change is the same every single time, you've found a simple pattern rule.
Mathematical Framework
Once you identify the type of pattern, you can write a formula. A formula lets you jump straight to any term without listing every term before it. Here are the two most common formulas you'll need for the SHSAT.
Let's break this down. Suppose the first term is 5 and you add 4 each time. To find the 10th term, you plug in: a₁ = 5, n = 10, d = 4. That gives you 5 + (10 − 1) × 4 = 5 + 36 = 41. Easy!
For a geometric sequence starting at 2 with a common ratio of 3, the 5th term is 2 × 3⁴ = 2 × 81 = 162. Notice the exponent is one less than the position number because the first term doesn't get multiplied at all.
Types of Pattern Rules
Not every sequence on the SHSAT is a simple add-or-multiply pattern. The chart below sorts the most common types you'll see, along with a quick way to identify each one.
| Pattern Type | Example | How to Spot It |
|---|---|---|
| Arithmetic (add/subtract) | 4, 7, 10, 13, 16, … | The difference between consecutive terms is always the same. |
| Geometric (multiply/divide) | 3, 12, 48, 192, … | The ratio between consecutive terms is always the same. |
| Alternating | 1, −2, 4, −8, 16, … | Signs flip between positive and negative. Often geometric with a negative ratio. |
| Growing difference | 1, 4, 9, 16, 25, … | Differences between terms increase by a steady amount. These are often perfect squares or other formulas. |
| Two-operation | 2, 5, 11, 23, 47, … | Each term uses two steps, like multiply by 2 then add 1. Check if a single operation doesn't work. |
Worked Example
Let's walk through a full SHSAT-style problem step by step.
Strategies: Strengths & Limitations
There are two main strategies for finding a term in a sequence: listing out every term one by one, or using a formula. Both work, but each has pros and cons.
| Strategy | Strengths | Limitations |
|---|---|---|
| Listing / Counting | Easy to understand. Works for any pattern. You don't need to memorize formulas. | Very slow for large positions (like the 50th term). Easy to make an arithmetic mistake along the way. |
| Using a Formula | Lightning fast for large positions. One calculation gives you any term. Saves time on a timed test like the SHSAT. | You need to correctly identify the type first. Plugging in the wrong values gives a wrong answer. Doesn't always exist for tricky patterns. |
Connection to Functions & Algebra
Sequences are actually connected to a bigger math idea: functions. When you write aₙ = 7 + (n − 1) × 4, you can think of that as a function where the input is the position number n and the output is the term. In high school algebra, you'll see this written as f(n) = 4n + 3.
| Concept Now | How It Grows Later |
|---|---|
| Arithmetic sequence rule (add a constant) | Linear function y = mx + b, where m is the common difference |
| Geometric sequence rule (multiply by a constant) | Exponential function y = a × rˣ, used in population growth and compound interest |
| Growing-difference pattern (like perfect squares) | Quadratic function y = ax² + bx + c, used in projectile motion |
| Finding the nth term | Evaluating a function at a specific input, f(n) |
Mastering pattern rules now gives you a head start on algebra and functions. Every time you find a pattern rule, you're actually building the kind of thinking that makes advanced math much easier.
Practice Problems
Lesson Summary
A sequence is an ordered list of numbers, and a pattern rule is the instruction that connects one term to the next. To identify the rule, start by finding the differences between terms. If the difference is always the same, you have an arithmetic sequence and can use the formula aₙ = a₁ + (n − 1) × d. If the differences aren't constant, divide consecutive terms to check for a common ratio. A constant ratio means you have a geometric sequence and can use aₙ = a₁ × r⁽ⁿ⁻¹⁾.
If neither a constant difference nor a constant ratio appears, look for two-step rules (like multiply then add) or growing differences (which point to squares or other formulas). Always verify your rule by testing it on terms you already know. These skills connect directly to functions in algebra and will help you solve sequence questions on the SHSAT quickly and confidently.