Missing Pattern Terms

Learn to identify the hidden rule behind a number pattern and find any missing term with confidence.

SSAT Middle Level: Quantitative

What Is a Number Pattern?

A number pattern (or sequence) is a list of numbers that follow a consistent rule. Each number in the list is called a term.

Why Patterns Matter on the SSAT

Pattern questions appear frequently on the SSAT Quantitative section. They test your ability to spot relationships, think logically, and work quickly.

Recognize

Identify the operation that connects consecutive terms.

Apply

Use the rule to extend the pattern forward or backward.

Verify

Check that your answer satisfies the rule for every pair of terms.

Common Pattern Rules

Arithmetic

Add or subtract the same number each time: \( 5, 8, 11, 14, \ldots \) (add \(3\))

Geometric

Multiply or divide by the same number each time: \( 2, 6, 18, 54, \ldots \) (multiply by \(3\))

Two-Step

Combine two operations: \( 1, 3, 7, 15, \ldots \) (multiply by \(2\), then add \(1\))

Alternating

The rule switches: \( 2, 6, 4, 8, 6, \ldots \) (add \(4\), then subtract \(2\))

The Core Rule for Finding Missing Terms

Arithmetic vs Geometric

	Arithmetic	Geometric
Rule	Add or subtract a constant \( d \)	Multiply or divide by a constant \( r \)
Formula	\( a_n = a_1 + (n-1)d \)	\( a_n = a_1 \times r^{\,n-1} \)
Example	\(5, 9, 13, 17\)	\(3, 12, 48, 192\)
Check	Differences are equal	Ratios are equal
Common on SSAT?	Very common	Common

When Differences Change

Sometimes the differences between terms are not constant: they form their own pattern. Look at the second differences.

How to Find the Missing Term

1

List the terms. Write down every given number in order.

2

Compute differences. Subtract each term from the next: \( a_{n+1} - a_n \).

3

Spot the rule. Are differences constant? Try ratios. Still not constant? Check second differences.

4

Apply the rule to find the missing term.

5

Verify. Plug the answer back in and confirm the pattern holds for every pair.

Example 1: Arithmetic Pattern

Find the missing term: \( 7, \; 13, \; ?, \; 25, \; 31 \)

Example 2: Geometric Pattern

Find the missing term: \( 2, \; 6, \; 18, \; ?, \; 162 \)

Example 3: Two-Step Pattern

Find the missing term: \( 1, \; 4, \; 13, \; ?, \; 121 \)

Example 4: Decreasing Pattern

Find the missing term: \( 100, \; 85, \; ?, \; 55, \; 40 \)

Patterns with Fractions and Negatives

The same strategies work with fractions and negative numbers. Always check both differences and ratios.

Fraction Example

\( 1, \; \frac{1}{2}, \; \frac{1}{4}, \; ?, \; \frac{1}{16} \)

Ratio: \( \times \frac{1}{2} \)
Missing: \( \frac{1}{8} \)

Negative Example

\( 10, \; 4, \; -2, \; ?, \; -14 \)

Difference: \( -6 \)
Missing: \( -8 \)

Common Mistakes to Avoid

✕ Assuming Addition

Jumping to "add a number" without checking if the pattern is multiplicative or two-step.

✕ Skipping Verification

Finding an answer but not plugging it back in to confirm the rule holds for every pair.

✕ Sign Errors

Miscalculating with negative differences or forgetting that subtraction changes direction.

Summary of Pattern Types

Type	Rule	How to Detect	Example
Arithmetic	\( +d \) or \( -d \)	Constant 1st diff	\(3, 7, 11, 15\)
Geometric	\( \times r \) or \( \div r \)	Constant ratio	\(5, 15, 45, 135\)
Two-Step	\( \times r + c \)	Neither works alone	\(1, 4, 13, 40\)
Quadratic	\( n^2 \) based	Constant 2nd diff	\(1, 4, 9, 16\)
Alternating	Two rules alternate	Odd/even positions differ	\(2, 6, 4, 8, 6\)

Question 1 of 5

What is the missing term in the sequence
\( 5, \; 11, \; 17, \; ?, \; 29 \)?

The common difference is \( 11 - 5 = 6 \). So the missing term is \( 17 + 6 = 23 \). Verify: \( 29 - 23 = 6 \) ✓

Question 2 of 5

What is the missing term in the sequence
\( 3, \; 9, \; 27, \; ?, \; 243 \)?

This is a geometric pattern with ratio \( \frac{9}{3} = 3 \). The missing term is \( 27 \times 3 = 81 \). Check: \( 81 \times 3 = 243 \) ✓

Question 3 of 5

What is the missing term in the sequence
\( 2, \; 5, \; 11, \; ?, \; 47 \)?

The rule is: multiply by \( 2 \), then add \( 1 \). Check: \( 2 \times 2 + 1 = 5 \), \( 5 \times 2 + 1 = 11 \), \( 11 \times 2 + 1 = 23 \). Verify: \( 23 \times 2 + 1 = 47 \) ✓

Question 4 of 5

What is the missing term in the sequence
\( 64, \; 32, \; 16, \; ?, \; 4 \)?

Each term is divided by \( 2 \) (or multiplied by \( \frac{1}{2} \)). The missing term is \( 16 \div 2 = 8 \). Check: \( 8 \div 2 = 4 \) ✓

Question 5 of 5

What is the missing term in the sequence
\( 1, \; 4, \; 9, \; 16, \; ?, \; 36 \)?

These are perfect squares: \( 1^2, 2^2, 3^2, 4^2, ?, 6^2 \). The missing term is \( 5^2 = 25 \). The second differences are constant at \( +2 \), confirming a quadratic pattern.