Pattern Rules From Sequences — Identify a pattern rule from a sequence. - SHSAT Math

SECTION 1

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.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid studied number patterns and wrote about sequences of even numbers, odd numbers, and perfect squares.

~600 CE • Indian Mathematics

Indian scholars like Aryabhata explored arithmetic sequences and used pattern rules to build tables for astronomy.

1202 • Fibonacci's Liber Abaci

Leonardo Fibonacci introduced a famous sequence (1, 1, 2, 3, 5, 8, …) where each term is the sum of the two before it.

1700s • Gauss & Arithmetic Series

As a young student, Carl Friedrich Gauss quickly added the numbers 1 through 100 by spotting a pattern rule. This story shows the power of recognizing sequences.

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?

SECTION 2

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

An ordered list of numbers, like 3, 6, 9, 12, … The three dots (called an ellipsis) mean the pattern continues.

Term

Each value in the sequence. In 3, 6, 9, 12, the first term is 3, the second term is 6, and so on.

Common Difference

The amount you add (or subtract) each time in an arithmetic sequence. For 3, 6, 9, 12 the common difference is 3.

Common Ratio

The number you multiply by each time in a geometric sequence. For 2, 6, 18, 54 the common ratio is 3.

Position Number (n)

The spot a term sits in. The 1st term has position n = 1, the 2nd has n = 2, etc. This helps you write a formula.

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KEY TAKEAWAY

Think of a sequence like a playlist on shuffle that actually has a secret order. The pattern rule is like the DJ's plan — once you figure it out, you know exactly which song (or number) comes next, even if you haven't heard it yet.

SECTION 3

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.

Left: the arithmetic sequence 5, 9, 13, 17 adds 4 each time. Right: the geometric sequence 2, 6, 18, 54 multiplies by 3 each time. The arrows show the operation that connects each term to the next.

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.

SECTION 4

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.

ARITHMETIC SEQUENCE FORMULA

aₙ = a₁ + (n − 1) × d

aₙ = the term you want to find, a₁ = the first term, n = the position number, d = the common difference (what you add each time).

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!

GEOMETRIC SEQUENCE FORMULA

aₙ = a₁ × r⁽ⁿ⁻¹⁾

aₙ = the term you want, a₁ = the first term, n = the position number, r = the common ratio (what you multiply by each time).

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.

FINDING THE COMMON DIFFERENCE

d = second term − first term

Subtract any term from the one right after it. If the answer is the same every time, you have an arithmetic sequence.

FINDING THE COMMON RATIO

r = second term ÷ first term

Divide any term by the one right before it. If the answer is the same every time, you have a geometric sequence.

SECTION 5

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.

Common sequence types on the SHSAT

Follow this flowchart when you see a sequence. First check for a common difference (arithmetic). If that doesn't work, check for a common ratio (geometric). If neither works, look for a two-step rule or growing differences.

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SHSAT TIP

Always write the differences between terms above the sequence. If those differences aren't equal, write the differences of the differences. This "second difference" trick reveals patterns like perfect squares (1, 4, 9, 16 …) very quickly.

SECTION 6

Worked Example

Let's walk through a full SHSAT-style problem step by step.

Find the 20th term of the sequence: 7, 11, 15, 19, 23, …

1
Step 1 — Find the differences

Subtract each term from the one after it. 11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4, 23 − 19 = 4. The difference is always 4.

Common difference d = 4
2
Step 2 — Identify the type

Because we add the same number every time, this is an arithmetic sequence.
3
Step 3 — Write down what you know

First term a₁ = 7, common difference d = 4, position we want n = 20.
4
Step 4 — Plug into the formula

Use aₙ = a₁ + (n − 1) × d. Substitute: a₂₀ = 7 + (20 − 1) × 4.
5
Step 5 — Simplify step by step

First, 20 − 1 = 19. Then, 19 × 4 = 76. Finally, 7 + 76 = 83.

The 20th term is 83.

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CHECK YOUR WORK

A great habit is to test your formula on a term you already know. Plug in n = 1: 7 + (1 − 1) × 4 = 7. That matches! Plug in n = 3: 7 + (3 − 1) × 4 = 7 + 8 = 15. That matches too. If the formula works for known terms, it will work for any term.

SECTION 7

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.

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BEST PRACTICE

On the SHSAT, use listing for small positions (like the 6th or 7th term) and the formula for large positions (like the 25th or 100th term). Think of it like walking versus driving: for a short trip across the street, walking is fine. For a long trip across the city, you definitely want a car (the formula).

SECTION 8

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)

From sequences now to functions later

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A sequence has the terms 10, 10, 10, 10, … Is this an arithmetic sequence, a geometric sequence, both, or neither? Explain your reasoning.

Show answer

This sequence is both arithmetic and geometric. It is arithmetic because the common difference is 0 (you add 0 each time: 10 + 0 = 10). It is also geometric because the common ratio is 1 (you multiply by 1 each time: 10 × 1 = 10). A constant sequence is a special case that satisfies both definitions.

Question 2: BASIC CALCULATION

Find the 15th term of the arithmetic sequence: 3, 8, 13, 18, 23, …

Show answer

First, find the common difference: 8 − 3 = 5. The first term is a₁ = 3, d = 5, and n = 15. Plug into the formula: a₁₅ = 3 + (15 − 1) × 5 = 3 + 14 × 5 = 3 + 70 = 73. You can check by noting the 2nd term: 3 + (2 − 1) × 5 = 8. ✓

Question 3: INTERMEDIATE

A sequence starts: 2, 6, 18, 54, … What is the 7th term? Identify the pattern type and use the correct formula.

Show answer

Check the differences: 6 − 2 = 4, 18 − 6 = 12. The differences are not equal, so it is not arithmetic. Check the ratios: 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3. The ratio is always 3, so this is a geometric sequence. Use aₙ = a₁ × r⁽ⁿ⁻¹⁾ with a₁ = 2, r = 3, n = 7. That gives 2 × 3⁶ = 2 × 729 = 1,458.

Question 4: APPLIED

Maya saves money each week. In week 1 she saves $5, in week 2 she saves $8, in week 3 she saves $11, and so on. How much will she save in week 12? How much total will she have saved after 12 weeks?

Show answer

The savings form an arithmetic sequence: 5, 8, 11, … with d = 3. Week 12 savings: a₁₂ = 5 + (12 − 1) × 3 = 5 + 33 = $38. For the total, add all 12 terms. The sum formula for an arithmetic sequence is S = n ÷ 2 × (first term + last term). So S = 12 ÷ 2 × (5 + 38) = 6 × 43 = $258 total saved after 12 weeks.

Question 5: CRITICAL THINKING

Consider the sequence: 2, 5, 11, 23, 47, … This is not arithmetic or geometric. Find the pattern rule and predict the 7th term.

Show answer

Check differences: 3, 6, 12, 24 — they double each time but aren't constant, so it's not arithmetic. Check ratios: 5 ÷ 2 = 2.5, 11 ÷ 5 = 2.2 — not constant, so it's not geometric. Try a two-step rule: 2 × 2 + 1 = 5 ✓, 5 × 2 + 1 = 11 ✓, 11 × 2 + 1 = 23 ✓, 23 × 2 + 1 = 47 ✓. The rule is "multiply by 2, then add 1." Continuing: 47 × 2 + 1 = 95 (6th term), 95 × 2 + 1 = 191 (7th term). This type of two-operation pattern is a favorite tricky question on the SHSAT.

SUMMARY

Lesson Summary

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.