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Numerical Sequence
Patterns

Master the art of recognizing and extending number patterns: from arithmetic and geometric sequences to more complex rules tested on the SSAT.

SSAT Upper Level

What Is a Sequence?

A sequence is an ordered list of numbers that follows a specific rule or pattern. Each number in the list is called a term.

2 Term 1 5 Term 2 8 Term 3 11 Term 4 ? Term 5
\(+3\)
\(+3\)
\(+3\)
\(+3\)

Key Vocabulary

Term
Each individual number in the sequence, often written as \( a_n \).
Common Difference
The constant value \( d \) added between consecutive terms in an arithmetic sequence.
Common Ratio
The constant factor \( r \) multiplied between consecutive terms in a geometric sequence.

Why Patterns Matter

The SSAT tests your ability to spot structure in numbers quickly. Pattern recognition lets you:

1
Predict missing terms without computing every value from scratch.
2
Solve problems faster by recognizing a known sequence type immediately.
3
Avoid traps where answer choices exploit common misreadings of the pattern.

Arithmetic Sequences

An arithmetic sequence adds (or subtracts) the same value \( d \) every step. The general formula for the \( n \)-th term is:

3 7 11 15 19
\(+4\)
\(+4\)
\(+4\)
\(+4\)
 \(a_1\) \(a_2\) \(a_3\) \(a_4\) \(a_5\)

Geometric Sequences

A geometric sequence multiplies (or divides) by the same factor \( r \) every step. The general formula is:

2 6 18 54 162
\(\times 3\)
\(\times 3\)
\(\times 3\)
\(\times 3\)

Arithmetic vs Geometric

Arithmetic Geometric
Operation Add or subtract Multiply or divide
Constant \( d \) (difference) \( r \) (ratio)
Formula \( a_1 + (n-1)d \) \( a_1 \cdot r^{n-1} \)
Growth Linear (steady) Exponential (fast)
Example 5, 8, 11, 14, ... 5, 10, 20, 40, ...

Beyond the Basics

Not every SSAT sequence is purely arithmetic or geometric. Watch for these advanced patterns:

Alternating
Signs or operations switch: \( +, -, +, - \ldots \) or two interleaved rules.
Quadratic
Differences between terms grow linearly: 1, 4, 9, 16, 25 ... (\( n^2 \) pattern).
Fibonacci-type
Each term is the sum of the two before it: 1, 1, 2, 3, 5, 8 ...

Step-by-Step Strategy

1
Write the differences. Subtract each term from the next: \( a_2 - a_1,\; a_3 - a_2, \ldots \)
2
Check for a constant difference. If all differences are the same, it is arithmetic.
3
Check the ratios. Divide each term by the previous one. Equal ratios mean geometric.
4
Look at second differences. If first differences are not constant but their differences are, the pattern is quadratic.
5
Apply the rule to predict the next term or fill in a blank.

Example 1: Arithmetic

Find the next term: 4, 11, 18, 25, ?

4 11 18 25 32
\(+7\)
\(+7\)
\(+7\)
\(+7\)

Example 2: Geometric

Find the next term: 5, 15, 45, 135, ?

5 15 45 135 405
\(\times 3\)
\(\times 3\)
\(\times 3\)
\(\times 3\)

Example 3: Quadratic Pattern

Find the next term: 2, 5, 10, 17, 26, ?

2 5 10 17 26 37 1st diff:
\(3\)
\(5\)
\(7\)
\(9\)
\(11\)
 2nd diff:
\(2\)
\(2\)
\(2\)
\(2\)
 Constant 2nd differences = quadratic pattern

Example 4: Alternating Pattern

Find the next term: 1, 4, 2, 8, 3, ?

1 2 3 4 8 12 odd even odd even odd even
Odd positions: 1, 2, 3, ... (\(+1\)) | Even positions: 4, 8, ... (\(+4\))

Common Mistakes

✕ Assuming arithmetic
Seeing a growing sequence and assuming it adds a constant without checking ratios or second differences.
✕ Ignoring negatives
Forgetting that \( d \) or \( r \) can be negative. The sequence 10, 7, 4, 1 has \( d = -3 \).
✕ Off-by-one in the formula
Using \( n \) instead of \( n - 1 \) in \( a_n = a_1 + nd \), which shifts every answer by one step.

Pattern Cheat Sheet

Pattern Type How to Identify Next-Term Rule
Arithmetic Constant 1st differences \( a_n + d \)
Geometric Constant ratios \( a_n \times r \)
Quadratic Constant 2nd differences Add next 1st diff
Alternating Two interleaved patterns Apply sub-rule
Fibonacci-type Term = sum of previous two \( a_{n-1} + a_{n-2} \)

Question 1 of 5

What is the next term in the sequence 6, 10, 14, 18, ...?

Question 2 of 5

What is the next term in the sequence 3, 9, 27, 81, ...?

Question 3 of 5

The sequence 1, 4, 9, 16, 25, ... follows which pattern?

Question 4 of 5

What is the 8th term of the arithmetic sequence whose first term is \( 5 \) and common difference is \( -3 \)?

Question 5 of 5

Find the next term: 2, 6, 3, 9, 4, ...