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CogAT (Grades 3+) • Quantitative Battery

Select the Number That Continues the Series

Learn to spot hidden patterns in number sequences and predict what comes next.

Explore the lesson 10 min read
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Where Do Number Patterns Come From?

Humans have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that some numbers follow a rule, like counting by twos (2, 4, 6, 8…). They realized that understanding these rules could help solve real-world problems, from building pyramids to predicting the seasons.

Today, the CogAT (Cognitive Abilities Test) uses number series questions to measure how well you can spot hidden rules. These questions don't test what you've memorized. Instead, they test your ability to think logically and find patterns.

~300 BC • Euclid's Elements

The Greek mathematician Euclid studied number patterns and wrote about sequences that grow by adding the same amount each time.

1202 • Fibonacci's Famous Sequence

Leonardo Fibonacci introduced a famous sequence (1, 1, 2, 3, 5, 8…) where each number is the sum of the two before it. This pattern appears everywhere in nature.

1800s • Sequences in Algebra

Mathematicians began writing formal rules (formulas) for sequences. This helped engineers and scientists make predictions.

1968 • The CogAT Is Created

Researchers developed the Cognitive Abilities Test. Number series questions became a key part of the Quantitative Battery to measure reasoning skills.

So here's the big question these problems ask: If you see a list of numbers that follow a hidden rule, can you figure out that rule and predict the next number? Let's learn how.

SECTION 2

Core Principles of Number Series

Every number series follows a rule (also called a pattern). Your job is to discover the rule by looking at how the numbers change from one to the next. Here are the main types of rules you'll see.

1

Addition / Subtraction Rule

The same number is added or subtracted each time. Example: 3, 7, 11, 15 → the rule is add 4.
2

Multiplication / Division Rule

Each number is multiplied or divided by the same value. Example: 2, 6, 18, 54 → the rule is multiply by 3.
3

Changing Difference Rule

The amount added changes in its own pattern. Example: 1, 2, 4, 7, 11 → add 1, add 2, add 3, add 4. The differences grow.
4

Alternating Rule

Two different operations take turns. Example: 2, 6, 4, 8, 6 → add 4, then subtract 2, repeating.
5

Two-Operation Rule

A combination of operations applies each step. Example: 1, 3, 7, 15 → the rule is multiply by 2, then add 1.
✦

KEY TAKEAWAY

Think of a number series like a playlist that follows a rule. If a DJ plays a song every 5 minutes, you can predict when the next song starts. Number series work the same way — once you find the rule, you can predict the next number!

SECTION 3

Seeing the Pattern

One of the best ways to understand a number series is to look at the gaps between numbers. The diagram below shows how to find the differences between each pair of numbers. These differences reveal the hidden rule.

Finding the Differences in a Number SeriesSeries: 5, 8, 11, 14, ?581114?+3+3+3+3RULE FOUND: Add 3 each time14 + 3 = 17 → The answer is 17Step 1: Find the difference between each pair of numbers.Step 2: Notice all differences are the same (+3).Step 3: Apply the rule to the last number to find the answer.
This diagram shows the series 5, 8, 11, 14, ?. The curved arrows represent the difference between each pair. Since every difference is +3, the rule is 'add 3,' and the missing number is 17.

Notice how we wrote the differences between each pair of numbers. When all the differences are the same, you've found a simple constant difference pattern. This is the most common type of number series you'll see on the CogAT.

SECTION 4

The Math Behind Number Series

You don't need fancy algebra to solve these problems! But knowing a few simple formulas can make you faster. Let's look at the math behind the two most common series types.

Arithmetic Series (Add or Subtract)

ARITHMETIC SERIES RULE

NextNumber=LastNumber+dNext Number = Last Number + d

Here, d stands for the common difference — the amount added (or subtracted) each step. For the series 5, 8, 11, 14, the value of d is 3.

In an arithmetic series (say "air-ith-MET-ik"), the difference between any two neighboring numbers is always the same. If you subtract the first number from the second, you get d. Subtract the second from the third, you get d again.

Geometric Series (Multiply or Divide)

GEOMETRIC SERIES RULE

NextNumber=LastNumber×rNext Number = Last Number × r

Here, r stands for the common ratio — the number you multiply by each step. For the series 3, 6, 12, 24, the value of r is 2.

In a geometric series, you multiply by the same number each time. If adding doesn't give you a constant difference, try dividing each number by the one before it. If you get the same answer every time, you've found a geometric pattern.

What If the Differences Change?

CHANGING DIFFERENCE

Differences:d1,d2,d3…→findthepatterninthedifferencesthemselvesDifferences: d₁, d₂, d₃ … → find the pattern in the differences themselves

For the series 2, 3, 5, 8, 12, the differences are 1, 2, 3, 4. The differences grow by 1 each time. So the next difference would be 5, making the next number 12 + 5 = 17.

💡

Pro Tip

If the first set of differences isn't constant, find the differences of the differences. This "second level" often reveals the hidden rule!

SECTION 5

A Closer Look at Pattern Types

The diagram below organizes the five main pattern types you might see on the CogAT. Use it like a decision tree: start at the top and follow the questions to figure out which pattern you're dealing with.

Pattern Type Decision TreeFind the differencesSame each time?Not the same?YES → Arithmetice.g. 4, 7, 10, 13 (+3)NO → Try ratiosDivide each by the one beforeSame ratio?Not same ratio?YES → Geometrice.g. 2, 6, 18, 54 (×3)NO → Go deeperCheck diff of differencesChanging Differencee.g. 1, 2, 4, 7, 11 (diffs: 1,2,3,4)Alternating / Two-OpLook for repeating patternsTip: Always start by finding the differences!Most CogAT series questions use arithmetic or changing-difference patterns.
This decision tree guides you through identifying the pattern type. Start by finding the differences. If they are constant, it's arithmetic. If not, check for a constant ratio (geometric), then try changing differences or alternating rules.
Pattern TypeExample SeriesRuleNext Number
Arithmetic (add)10, 15, 20, 25Add 530
Arithmetic (subtract)50, 43, 36, 29Subtract 722
Geometric (multiply)4, 12, 36, 108Multiply by 3324
Changing difference3, 4, 6, 9, 13Add 1, 2, 3, 4…18
Alternating5, 10, 7, 12, 9+5, −3, +5, −3…14

Examples of each number series pattern type

SECTION 6

Worked Example: Step by Step

Let's walk through a tricky example together. This one uses a changing difference pattern.

ℹ

The Problem

What number comes next in the series: 2, 5, 10, 17, 26, ?

Solving a Changing-Difference Series

  1. 1

    Step 1 — Find the first-level differences

    Subtract each number from the one after it. 5 − 2 = 3, 10 − 5 = 5, 17 − 10 = 7, 26 − 17 = 9.
    Differences: 3, 5, 7, 9
  2. 2

    Step 2 — Check if the differences are constant

    The differences are 3, 5, 7, 9. These are NOT all the same, so this is NOT a simple arithmetic series. Let's go deeper.
    Not constant — try second-level differences
  3. 3

    Step 3 — Find the second-level differences

    Now find the differences of the differences: 5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2.
    Second-level differences: 2, 2, 2 (constant!)
  4. 4

    Step 4 — Predict the next first-level difference

    Since the second-level difference is always 2, the next first-level difference is 9 + 2 = 11.
    Next difference: 11
  5. 5

    Step 5 — Find the next number in the series

    Add the predicted difference to the last number in the series: 26 + 11 = 37.
    Answer: 37
✦

KEY TAKEAWAY

Think of it like climbing stairs. In a simple pattern, each step is the same height. In a changing-difference pattern, each step gets a little taller (or shorter). By measuring how much taller each step gets, you can predict the next one!

SECTION 7

Strategies, Strengths & Common Mistakes

Knowing the pattern types is important, but you also need smart test-taking strategies. The table below shows what works well and what mistakes to avoid.

Strategy ✓Common Mistake ✗
Write down the differences between each pair of numbers before guessing.Jumping to an answer without checking the differences.
If the differences aren't constant, try dividing (look for a ratio).Only trying addition or subtraction and giving up.
Check your answer by seeing if it fits the pattern going backward.Picking an answer without checking it against the whole series.
Consider alternating rules when differences seem to jump around.Assuming every series uses only one operation.
Use the answer choices to help — try each one and see which fits.Spending too much time on one problem instead of moving on.

Effective strategies vs. common mistakes on number series questions

🎯

REMEMBER

On the CogAT, you'll have answer choices to choose from. If you get stuck, try plugging each answer choice back into the series to see which one keeps the pattern going. It's like trying keys in a lock — the right one will click!

SECTION 8

Beyond the Basics: Harder Patterns

Once you master the basic patterns, you might encounter trickier series. The table below compares the standard patterns to their harder versions that sometimes appear on more challenging questions.

Standard PatternAdvanced VersionExample
Add the same numberAdd a number that changes in a pattern1, 3, 6, 10, 15 (add 2, 3, 4, 5)
Multiply by the same numberMultiply then add (two-operation)1, 3, 7, 15 (×2 then +1)
Alternating two operationsAlternating operations with changing values2, 4, 3, 6, 4 (+2, −1, +3, −2)
Perfect squaresSquares plus or minus a constant2, 5, 10, 17 (1²+1, 2²+1, 3²+1, 4²+1)

Standard vs. advanced number series patterns

Don't worry if these harder patterns seem tough right now. As you practice, you'll start to recognize them faster. The key is to always start with the differences and work your way through the decision tree. Advanced series just have one more layer to peel back.

📝

Looking Ahead

In high school, you'll learn about these patterns in algebra class using formulas with variables like aₙ = a₁ + (n − 1)d. For now, the difference method you've learned here is all you need!

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. For each one, find the differences first, then determine the rule. Check your answer against the solution when you're done.

Question 1: CONCEPTUAL

What number comes next? 6, 10, 14, 18, ?
Show answer
The answer is 22. The differences are 10 − 6 = 4, 14 − 10 = 4, 18 − 14 = 4. Every difference is 4, so this is an arithmetic series with d = 4. Add 4 to the last number: 18 + 4 = 22.

Question 2: BASIC CALCULATION

What number comes next? 3, 9, 27, 81, ?
Show answer
The answer is 243. The differences are 6, 18, 54, which aren't constant. Try dividing: 9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3. The common ratio is 3, so this is a geometric series. Multiply the last number by 3: 81 × 3 = 243.

Question 3: INTERMEDIATE

What number comes next? 4, 5, 7, 10, 14, ?
Show answer
The answer is 19. The first-level differences are 1, 2, 3, 4. These aren't constant, but they increase by 1 each time (second-level difference = 1). So the next first-level difference is 5, and the next number is 14 + 5 = 19. This is a changing-difference pattern.

Question 4: APPLIED

What number comes next? 100, 95, 85, 70, 50, ?
Show answer
The answer is 25. The differences are −5, −10, −15, −20. Each difference decreases by 5 (second-level difference = −5). The next difference is −25, so the next number is 50 + (−25) = 25. This is a changing-difference pattern where the numbers get smaller faster and faster.

Question 5: CRITICAL THINKING

What number comes next? 2, 6, 4, 12, 10, 30, ?
Show answer
The answer is 28. This series alternates between two operations. Starting from 2: multiply by 3 (2 × 3 = 6), subtract 2 (6 − 2 = 4), multiply by 3 (4 × 3 = 12), subtract 2 (12 − 2 = 10), multiply by 3 (10 × 3 = 30). The next operation is subtract 2, so 30 − 2 = 28. Recognizing alternating rules is the key to cracking this type of problem.

SUMMARY

Summary: Cracking Number Series

Summary

Number series questions on the CogAT ask you to find the hidden rule behind a list of numbers. The five main pattern types are arithmetic (constant addition or subtraction), geometric (constant multiplication or division), changing difference, alternating, and two-operation patterns.

Your go-to strategy is to find the differences between each pair of numbers. If those differences are the same, you have an arithmetic series. If not, try ratios for a geometric series, or look at the differences of the differences for a changing pattern. Always check your answer by making sure it continues the rule. With practice, spotting these patterns becomes fast and natural!

Varsity Tutors • CogAT (Grades 3+) • Select the Number That Continues the Series