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SSAT Middle Level: Quantitative • Patterns and Sequences

Find a Missing Term in a Number Pattern

Learn to spot the hidden rule in a sequence and use it to fill in any missing number.

Explore the lesson 8 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 9 sections 5 questions 6 figures

SECTION 1

Where Do Number Patterns Come From?

People have been fascinated by number patterns (sequences of numbers that follow a rule) for thousands of years. Ancient civilizations noticed that certain arrangements of numbers appear again and again in nature, music, and architecture. Understanding these patterns helped early mathematicians build calendars, predict harvests, and even design buildings. Today, finding missing terms in a pattern is a core skill tested on exams like the SSAT.

2000 BCE • Babylonian Sequences

Ancient Babylonians recorded lists of square numbers and used patterns to solve problems about land and trade.

300 BCE • Greek Figurate Numbers

Greek mathematicians like Pythagoras studied triangular and square numbers, arranging dots into shapes to discover rules.

1202 • Fibonacci's Rabbit Problem

Leonardo of Pisa (Fibonacci) described a famous sequence—1, 1, 2, 3, 5, 8, 13…—where each term is the sum of the two before it.

Today • Patterns on the SSAT

Standardized tests ask you to find a missing term in a sequence. This checks your ability to spot rules and think logically.

The big question is simple: if someone gives you a list of numbers with one number missing, how do you figure out what belongs in the gap? That is exactly what this lesson will teach you.

SECTION 2

Core Principles of Number Patterns

Before you can find a missing term, you need to understand a few key ideas. Every number pattern has a rule (a repeating operation that connects one term to the next). Your job is to discover that rule and then apply it.

1

Term

Each number in a sequence is called a term. The first number is the 1st term, the second number is the 2nd term, and so on.
2

Common Difference

In an arithmetic sequence (a pattern that adds or subtracts the same amount each time), that amount is called the common difference.
3

Common Ratio

In a geometric sequence (a pattern that multiplies or divides by the same amount each time), that amount is called the common ratio.
4

Check Your Rule

Always test your rule on at least two pairs of consecutive terms before using it. If it works for all known terms, you have the right rule.

SECTION 3

Seeing the Pattern

A picture can make a pattern much easier to understand. The diagram below shows an arithmetic sequence where each term increases by 4. Notice how the arrows between the terms all show the same jump of +4. When a term is missing, you simply apply that same jump to fill the gap.

Arithmetic Sequence: +4 Each Step31st term+472nd term+4?3rd term+4154th term+4195th termFinding the Missing 3rd TermMethod 1: 7 + 4 = 11 (add the common difference to the 2nd term)Method 2: 15 − 4 = 11 (subtract the common difference from the 4th term)
The dashed yellow box shows the missing 3rd term. Since every jump is +4, you can work forward from 7 (getting 11) or backward from 15 (also getting 11). Both methods give the same answer: 11.

Notice that you can approach a missing term from either direction. Working forward means adding the rule to the term before the gap. Working backward means doing the opposite operation on the term after the gap. Both should give the same answer, which is a great way to double-check your work.

SECTION 4

The Math Behind Patterns

You do not need fancy formulas for most SSAT pattern problems. However, knowing a couple of simple equations can speed things up and make you more confident.

COMMON DIFFERENCE

d=nextterm−currenttermd = next term − current term

Pick any two terms that sit next to each other. Subtract the earlier one from the later one. The result, d, is the common difference. If d is positive, the pattern goes up. If d is negative, the pattern goes down.

FIND ANY TERM (ARITHMETIC)

termn=firstterm+(n−1)×dterm_n = first term + (n − 1) × d

Here, n is the position number of the term you want, and d is the common difference. For example, to find the 5th term when the first term is 3 and d = 4: 3 + (5 − 1) × 4 = 3 + 16 = 19.

COMMON RATIO (GEOMETRIC)

r=nextterm÷currenttermr = next term ÷ current term

For patterns that multiply instead of add, divide any term by the one before it. The result, r, is the common ratio. For example, in the sequence 2, 6, 18, 54, the common ratio is 6 ÷ 2 = 3.

ℹ
Quick Tip

SECTION 5

Types of Patterns You Will See

Not every pattern works the same way. The diagram below shows four common types. Learning to recognize each type quickly will save you time on test day.

Four Common Pattern Types1. Add / Subtract (Arithmetic)5, 8, 11, 14, 17 …Rule: add 3 each time (d = +3)2. Multiply / Divide (Geometric)4, 12, 36, 108 …Rule: multiply by 3 each time (r = 3)3. Two-Step Rule1, 3, 7, 15, 31 …Rule: multiply by 2, then add 1(1×2+1=3, 3×2+1=7, 7×2+1=15 …)4. Changing Difference2, 3, 5, 8, 12, 17 …Differences: +1, +2, +3, +4, +5 …The difference itself grows by 1 each timeStrategy FlowchartStep 1Subtract pairsStep 2All same? → ArithmeticStep 3Not same? → Try dividingStep 4All same ratio? → GeometricStep 5Neither? → Check for changing diff or two-step rule
Start by subtracting consecutive terms. If the differences are the same, you have an arithmetic pattern. If not, try dividing. If neither approach gives a constant, look for a changing difference or a two-step rule.
Pattern TypeHow to Spot ItExample
ArithmeticDifferences between consecutive terms are the same.10, 15, 20, 25 (d = +5)
GeometricRatios between consecutive terms are the same.3, 9, 27, 81 (r = ×3)
Two-StepNeither differences nor ratios are constant; two operations are needed.2, 5, 11, 23 (×2 + 1)
Changing DifferenceDifferences form their own pattern (e.g., +1, +2, +3, +4).1, 2, 4, 7, 11 (diffs: 1, 2, 3, 4)

Summary of the four main pattern types you should know for the SSAT.

SECTION 6

Worked Example: Finding the Missing Term

Let's walk through a complete example step by step. Suppose you see this problem on the SSAT:

Worked example

  1. 1

    Step 1 — Check for an Arithmetic Pattern

    Subtract consecutive known terms to look for a common difference. From 3 to 7: 7 − 3 = 4. From 19 to 27: 27 − 19 = 8. The differences are not the same, so this is not a simple arithmetic pattern.
  2. 2

    Step 2 — Check for a Geometric Pattern

    Try dividing consecutive known terms: 7 ÷ 3 ≈ 2.33, and 27 ÷ 19 ≈ 1.42. Those ratios are not the same, so this is not a simple geometric pattern either.
  3. 3

    Step 3 — Look at the Differences More Carefully

    Let's list what the differences could be. From 3 to 7 the difference is 4. From 19 to 27 the difference is 8. Notice that 8 = 4 × 2. What if the differences themselves grow? If the differences are 4, 4+4=8? No — let's try differences that increase by a fixed amount: 4, 6, 8, 10? Check: 3+4=7 ✓, 7+6=13, 13+8=21 ✗ (not 19). Instead try increasing by 2 each step: differences 4, 6, 8. Then: 3+4=7 ✓, 7+6=13, 13+8=21 ✗. Let's reconsider. Try differences 4, 8, 4, 8 (alternating)? 3+4=7 ✓, 7+8=15, 15+4=19 ✓, 19+8=27 ✓. An alternating pattern of +4, +8, +4, +8 works!
  4. 4

    Step 4 — Test the Theory

    The rule alternates: add 4, then add 8, then add 4, then add 8. Start with 3. Add 4: 3 + 4 = 7 ✓. Add 8: 7 + 8 = 15 (this is the missing term). Add 4: 15 + 4 = 19 ✓. Add 8: 19 + 8 = 27 ✓. Every known term checks out!
  5. 5

    Step 5 — Double-Check from the Other Direction

    Working backward from 19: the step before 19 should be −4 (since the alternating rule means we subtracted 4 to get to 19 from the missing term). So 19 − 4 = 15. This confirms our answer: the missing term is 15.

SECTION 7

Common Mistakes to Avoid

Knowing the right strategies can help you solve pattern problems faster. But it's equally important to know about common mistakes so you can avoid them.

1

Only Checking One Pair

Always verify your rule against every pair of consecutive terms you can. A rule that works for one pair might fail for another.
2

Assuming Arithmetic First

Not all patterns add or subtract a fixed amount. If differences aren't constant, move on and try other pattern types before guessing.
3

Skipping the Double-Check

After finding your missing term, always verify by applying the rule in both directions. This catches arithmetic errors quickly.
4

Forgetting to Read Carefully

Make sure you know exactly which term is missing. Miscounting positions is a common source of errors on timed tests.

SECTION 8

Where This Skill Leads

Finding a missing term is your first step into a much bigger world. In later math courses, you'll study these ideas more formally using algebra and even calculus. Here is a quick peek at how the skills you're learning now connect to what comes next.

1

Algebra

In algebra, sequences are written with formulas. For example, the nth term of an arithmetic sequence is a + (n−1)d. You're already thinking this way when you identify d!
2

Geometry

Geometric sequences appear in the formulas for compound interest and exponential growth—key ideas in finance and science.
3

Data Analysis

Spotting patterns in data is the foundation of statistics. The pattern-recognition skills you're building now carry directly into analyzing graphs and tables.
4

Computer Science

Algorithms often rely on sequences and recurrence relations. Understanding how one term connects to the next is a core idea in programming and problem-solving.

For now, focus on mastering the basics: identify the rule, apply it, and check your answer. These habits will serve you well on the SSAT and in every math class you take from here on.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. For each one, find the missing term and pick the correct answer from the five choices.

Question 1: Problem 1

What is the missing term in the sequence: 5, 10, __, 20, 25?
  • A. 12
  • B. 14
  • C. 15
  • D. 16
  • E. 18
Show answer
The sequence adds 5 each time (arithmetic, d = +5). After 10, the next term is 10 + 5 = 15. Check: 15 + 5 = 20 ✓.

Question 2: Problem 2

What is the missing term in the sequence: 3, 6, 12, __, 48?
  • A. 18
  • B. 20
  • C. 24
  • D. 30
  • E. 36
Show answer
Each term is multiplied by 2 (geometric, r = 2). After 12, the next term is 12 × 2 = 24. Check: 24 × 2 = 48 ✓.

Question 3: Problem 3

What is the missing term in the sequence: 1, 2, 4, 7, __, 16?
  • A. 9
  • B. 10
  • C. 11
  • D. 12
  • E. 13
Show answer
The differences increase by 1 each time: +1, +2, +3, +4, +5. After 7, add 4 to get 11. Check: 11 + 5 = 16 ✓.

Question 4: Problem 4

What is the missing term in the sequence: 2, 5, 11, 23, __?
  • A. 35
  • B. 41
  • C. 44
  • D. 47
  • E. 50
Show answer
The rule is multiply by 2, then add 1. Check: 2×2+1=5 ✓, 5×2+1=11 ✓, 11×2+1=23 ✓, 23×2+1=47.

Question 5: Problem 5

What is the missing term in the sequence: 100, 50, 25, __, 6.25?
  • A. 10
  • B. 12
  • C. 12.5
  • D. 15
  • E. 20
Show answer
Each term is divided by 2 (geometric, r = ½). After 25, the next term is 25 ÷ 2 = 12.5. Check: 12.5 ÷ 2 = 6.25 ✓.

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