Identify Number
Sequences
Spot the rule behind a list of numbers and predict what comes next, like \( 2, 5, 8, 11, \, ? \)
What Is a Number Sequence?
A number sequence is an ordered list of numbers that follows a fixed rule. Each number is called a term. Your job is to find the rule that connects the terms.
Why Sequences Matter
On the HSPT, sequence questions test your ability to spot patterns quickly. Once you find the rule, you can extend a list, fill in a missing term, or predict any future term.
Arithmetic Sequences
In an arithmetic sequence, you add or subtract the same number each time. This number is called the common difference.
Geometric Sequences
In a geometric sequence, you multiply or divide by the same number each time. This number is called the common ratio.
The Core Rule
Most HSPT sequences are built from a repeating operation. To extend the list, apply that same operation to the last known term.
where \( d \) is the common difference.
where \( r \) is the common ratio.
Arithmetic vs Geometric
| Arithmetic | Geometric | |
|---|---|---|
| Operation | Add or subtract | Multiply or divide |
| Special name | Common difference \( d \) | Common ratio \( r \) |
| Example | \( 4, 9, 14, 19 \) | \( 4, 8, 16, 32 \) |
| Growth | Steady, even steps | Faster, doubling style |
Other Common Patterns
Not every sequence is purely arithmetic or geometric. Watch for these other patterns too:
add \( +1, +2, +3, +4 \)
these are \( n^2 \)
two patterns woven together
A Step-by-Step Approach
Example 1: Find the Next Term
Find the next term in \( 5, 11, 17, 23, \, ? \)
Example 2: A Geometric Jump
Find the next term in \( 3, 12, 48, 192, \, ? \)
Example 3: Growing Gaps
Find the missing term in \( 2, 3, 5, 8, \, ?, 17 \)
Common Mistakes
Quick Reference
| Pattern | How to Spot It | Example |
|---|---|---|
| Arithmetic | Equal differences | \( 6, 10, 14, 18 \) |
| Geometric | Equal ratios | \( 2, 6, 18, 54 \) |
| Growing gaps | Differences increase | \( 1, 2, 4, 7, 11 \) |
| Squares | \( 1, 4, 9, 16 \) shape | \( n^2 \) values |
Question 1 of 5
What is the next term in \( 4, 9, 14, 19, \, ? \)
Question 2 of 5
What is the next term in \( 5, 10, 20, 40, \, ? \)
Question 3 of 5
Find the missing term: \( 3, \, ?, 11, 15 \)
Question 4 of 5
What comes next in \( 1, 2, 4, 7, 11, \, ? \)
Question 5 of 5
What is the next term in \( 64, 32, 16, 8, \, ? \)
You Can Crack Any Sequence
You now have a reliable routine for HSPT sequence questions: find the gaps, test for equal differences or ratios, then look for deeper patterns when needed.