Recognize a pattern in a sequence of numbers - CogAT Grades 3 Plus

SECTION 1

Why Do Number Patterns Matter?

People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain numbers follow rules. These rules helped them solve problems in building, trading, and even music. Understanding patterns is one of the most important skills in math — and in everyday life.

On the CogAT Quantitative Battery, you will see a row of numbers and need to figure out the rule that connects them. Then you pick the number that comes next. Let's explore how pattern-finding has developed over time.

~500 BCE • Pythagoreans Study Number Patterns

Ancient Greek mathematicians discovered that numbers could form shapes — like triangular and square numbers — by arranging pebbles in patterns.

~800 CE • Arabic Scholars Advance Algebra

Mathematicians like Al-Khwarizmi used patterns in numbers to develop algebra, a tool for writing rules as equations.

1202 • Fibonacci Publishes His Famous Sequence

Leonardo of Pisa (Fibonacci) introduced the sequence 1, 1, 2, 3, 5, 8, 13… where each number is the sum of the two before it. This pattern appears in nature — in sunflowers, pinecones, and seashells.

1700s • Carl Friedrich Gauss — The Prince of Mathematicians

As a child, Gauss quickly added the numbers 1 to 100 by spotting a pattern: pairing numbers from opposite ends always gave the same sum (1 + 100 = 101, 2 + 99 = 101, and so on).

Today • Patterns Power Technology

Computer algorithms, weather prediction, and even social media feeds rely on recognizing patterns in data. The CogAT tests this same skill.

The big question that pattern recognition answers is simple: What comes next, and why? Once you can answer that, you've cracked the code behind the sequence.

SECTION 2

Core Principles of Number Patterns

Every number sequence follows a rule (a set of instructions that tells you how to get from one number to the next). Your job is to find that rule. Here are the core ideas you need.

Look at the Differences

Subtract each number from the one after it. If the differences are all the same, you have a constant difference pattern (also called an arithmetic sequence).

Look at the Ratios

Divide each number by the one before it. If the ratios are all the same, you have a constant ratio pattern (also called a geometric sequence).

Check for Changing Differences

Sometimes the differences themselves form a pattern. For example, the differences might go up by 1 each time (like +2, +3, +4…). This means the rule involves a growing step.

Watch for Alternating Rules

Some sequences switch between two rules. For instance, one step might add 3 and the next might subtract 1. Look at every other number to spot alternating patterns.

Use More Than One Operation

Tricky sequences may combine operations — like multiply by 2 then add 1. If simple differences or ratios don't work, try combining two operations.

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KEY TAKEAWAY

Think of a number pattern like a playlist on shuffle-repeat. Once you figure out the order the songs play in (the rule), you can predict exactly which song comes next. The 'songs' are the numbers, and the 'rule' is the pattern the playlist follows.

SECTION 3

Seeing Patterns: A Visual Guide

The diagram below shows three common types of number sequences. Each row shows the numbers, the operation between them, and how the pattern looks when you chart the differences or ratios. Study each row carefully.

Three pattern types: arithmetic (constant difference, shown in cyan), geometric (constant ratio, shown in violet), and growing difference (differences that change in a pattern, shown in amber). The highlighted number at the end of each row is the predicted next term.

Notice that each row has a different kind of rule. In the first row, you always add the same number. In the second row, you always multiply by the same number. In the third row, the amount you add keeps growing. When you see a sequence on a test, try each approach until one fits.

SECTION 4

The Math Behind Patterns

You don't need complicated formulas for the CogAT, but knowing a few shortcuts can help you check your work quickly. Here are the key formulas for the most common pattern types.

ARITHMETIC SEQUENCE (CONSTANT DIFFERENCE)

next term = current term + d

Here, d stands for the common difference — the number you add (or subtract) each time. For example, in 3, 8, 13, 18…, d = 5.

GEOMETRIC SEQUENCE (CONSTANT RATIO)

next term = current term × r

Here, r stands for the common ratio — the number you multiply by each time. For example, in 2, 4, 8, 16…, r = 2.

TWO-OPERATION RULE

next term = (current term × a) + b

Some patterns use two steps. First multiply by a, then add b. For example, in 1, 3, 7, 15…, the rule is: multiply by 2, then add 1.

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Test Tip

Always start by writing out the differences between neighboring numbers. If those differences are the same, you're done! If not, write out the differences of the differences. Most CogAT questions can be solved within two rounds of differences.

SECTION 5

A Closer Look at Pattern Types

Let's organize the most common pattern types you might encounter. The diagram below maps each type and gives you a quick example. After the diagram, a table summarizes how to spot each one.

This decision tree walks you through how to identify a pattern type. Start at the top by finding the differences. If they're the same, it's arithmetic. If not, check ratios (geometric) or second differences (growing difference). If those don't work, try alternating or two-operation rules.

Pattern Type	How to Spot It	Example
Arithmetic	Differences between terms are all the same.	4, 7, 10, 13 → d = +3
Geometric	Each term divided by the previous gives the same ratio.	5, 15, 45, 135 → r = ×3
Growing Difference	Differences increase (or decrease) by a fixed amount.	2, 3, 5, 8, 12 → +1, +2, +3, +4
Alternating	The rule switches between two operations.	1, 4, 2, 5, 3, 6 → +3, −2
Two-Operation	Each step uses multiply (or divide) then add (or subtract).	2, 5, 11, 23 → ×2 + 1

Summary of common number pattern types

SECTION 6

Worked Example: Finding the Pattern

Let's walk through a CogAT-style problem step by step. Suppose you see this sequence and need to find the missing number:

Sequence: 2, 6, 18, 54, ?

Finding the Next Term

1
Step 1 — Find the Differences

Subtract each term from the next: 6 − 2 = 4, 18 − 6 = 12, 54 − 18 = 36. The differences are 4, 12, 36. These are not the same, so this is not a simple arithmetic pattern.

Differences: 4, 12, 36 — not constant
2
Step 2 — Try Ratios Instead

Divide each term by the one before it: 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3. Every ratio is 3. That means this is a geometric sequence with r = 3.

Common ratio r = 3
3
Step 3 — Apply the Rule

Multiply the last known term by the ratio: 54 × 3 = 162.

The missing number is 162.
4
Step 4 — Check Your Answer

Read the sequence with 162 in place: 2, 6, 18, 54, 162. Each number is 3 times the one before it. ✓ The answer checks out!

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REMEMBER

When differences aren't constant, switch to checking ratios. It's like trying a different key on a lock — if one key (subtraction) doesn't work, try another (division) until the lock opens.

SECTION 7

Strengths & Limitations of Each Strategy

Not every strategy works for every pattern. The table below shows when each approach is strongest and when it might let you down.

Strategy	Strengths	Limitations
Check Differences	Quick and easy. Works for the most common CogAT sequences. Good first step every time.	Fails when the pattern uses multiplication, division, or alternating rules.
Check Ratios	Catches geometric patterns fast. Very reliable when numbers grow quickly.	Doesn't help if the sequence includes zero or if the rule mixes addition and multiplication.
Second Differences	Reveals growing-difference patterns like perfect squares or triangular numbers.	Requires more calculations, which can eat up time on a timed test.
Look at Every Other Term	Ideal for alternating patterns. Splits a confusing sequence into two simpler ones.	Only useful when there truly are two interleaved sub-patterns.

Comparing pattern-recognition strategies

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KEY TAKEAWAY

Think of these strategies as tools in a toolbox. A hammer is great for nails but not for screws. Similarly, checking differences is great for arithmetic patterns but not for geometric ones. Skilled problem-solvers try one tool, and if it doesn't fit, they calmly grab another.

SECTION 8

Connection to Advanced Thinking

The pattern skills you're learning now are stepping stones to bigger ideas in math and science. Here's a preview of where these concepts lead.

What You Learn Now	Where It Leads
Arithmetic sequences (constant difference)	Linear equations and graphing straight lines in algebra.
Geometric sequences (constant ratio)	Exponential growth and decay — used in biology, finance, and physics.
Growing differences (like perfect squares)	Quadratic equations and parabolas in advanced algebra.
Two-operation rules	Recursive functions in computer programming.

From CogAT patterns to advanced math

By getting comfortable with patterns now, you're building a foundation for high school algebra, computer science, and even data science. Every time you spot a pattern, you're thinking the way mathematicians and scientists do.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. For each one, find the pattern and predict the missing number. Check the answer after you've given it a try.

Question 1: CONCEPTUAL

What type of pattern is this: 10, 15, 20, 25, 30? Is it arithmetic, geometric, or something else? How do you know?

Show answer

This is an arithmetic pattern. You can tell because the difference between each pair of neighboring numbers is always the same: 15 − 10 = 5, 20 − 15 = 5, 25 − 20 = 5, and 30 − 25 = 5. The common difference is +5. Whenever the difference is constant, the sequence is arithmetic.

Question 2: BASIC CALCULATION

Find the next number: 4, 12, 36, 108, ?

Show answer

First, check ratios: 12 ÷ 4 = 3, 36 ÷ 12 = 3, 108 ÷ 36 = 3. The common ratio is ×3, so this is a geometric sequence. Multiply the last term by 3: 108 × 3 = 324. You can double-check: the full sequence is 4, 12, 36, 108, 324 — each number is three times the one before it.

Question 3: INTERMEDIATE

What is the missing number? 3, 4, 6, 9, 13, 18, ?

Show answer

Find the differences: 4 − 3 = 1, 6 − 4 = 2, 9 − 6 = 3, 13 − 9 = 4, 18 − 13 = 5. The differences are 1, 2, 3, 4, 5 — they increase by 1 each time. This is a growing-difference pattern. The next difference should be 6, so 18 + 6 = 24. The full sequence is 3, 4, 6, 9, 13, 18, 24.

Question 4: APPLIED

A baker made cookies each day for a week. On Monday she made 2, Tuesday she made 5, Wednesday she made 11, and Thursday she made 23. If the pattern continues, how many cookies does she make on Friday?

Show answer

Check differences: 5 − 2 = 3, 11 − 5 = 6, 23 − 11 = 12. The differences (3, 6, 12) are not constant, but their ratios are: 6 ÷ 3 = 2 and 12 ÷ 6 = 2. Let's try a two-operation rule instead. From 2 to 5: 2 × 2 + 1 = 5. From 5 to 11: 5 × 2 + 1 = 11. From 11 to 23: 11 × 2 + 1 = 23. The rule is ×2 + 1. For Friday: 23 × 2 + 1 = 47 cookies.

Question 5: CRITICAL THINKING

Look at this sequence: 1, 4, 2, 8, 3, 12, 4, ?. What is the missing number, and can you explain the two separate patterns hidden inside?

Show answer

This sequence alternates between two sub-patterns. Separate the odd-position terms (1st, 3rd, 5th, 7th): 1, 2, 3, 4 — this is a simple arithmetic sequence adding 1 each time. Now look at the even-position terms (2nd, 4th, 6th): 4, 8, 12 — this is an arithmetic sequence adding 4 each time. The missing 8th term belongs to the even-position group, so 12 + 4 = 16. The full sequence is 1, 4, 2, 8, 3, 12, 4, 16. Whenever a sequence looks confusing, try checking every other term — you might find two simpler patterns woven together.

SUMMARY

Lesson Summary

Summary

Recognizing number patterns means finding the hidden rule that connects terms in a sequence. Start by computing the differences between terms. If the differences are all the same, you have an arithmetic sequence. If not, check ratios for a geometric sequence, or look at second differences for growing-difference patterns. Some sequences use alternating rules or two-operation rules.

When you face a CogAT pattern question, follow the decision tree: try differences first, then ratios, then second differences, and finally look for alternating or combined operations. Always check your answer by reading the sequence with your predicted number in place. Pattern recognition is a skill that gets better with practice — the more sequences you try, the faster you'll spot the rule.