Identify Number Sequences - HSPT Quantitative

SECTION 1

Historical Context & Motivation

People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain numbers follow rules. Once you know the rule, you can predict the next number in the list. This skill is a big part of the HSPT Quantitative section.

A number sequence is simply a list of numbers arranged in a specific order. Each number in the list is called a term. The magic is in the pattern that connects one term to the next.

~300 BC • Euclid's Elements

The Greek mathematician Euclid studied sequences of numbers and wrote rules for finding perfect numbers.

~600 AD • Indian Mathematicians

Scholars in India explored arithmetic and geometric progressions, writing formulas to find sums of long lists.

1202 • Fibonacci's Liber Abaci

Leonardo Fibonacci introduced his famous sequence (1, 1, 2, 3, 5, 8, …) to Europe, where each term is the sum of the two before it.

1700s • Carl Friedrich Gauss

As a young student, Gauss quickly added the numbers 1 through 100 by spotting a pattern, amazing his teacher.

On the HSPT, you will see a list of numbers and need to figure out the rule behind the pattern. Then you pick the number that comes next. Let's learn how!

SECTION 2

Core Principles & Definitions

Before we dive into examples, let's nail down the key ideas you need. Every number sequence question on the HSPT boils down to finding a pattern. Here are the main types of sequences you will see.

Arithmetic Sequence

You add (or subtract) the same number each time. Example: 3, 7, 11, 15, … (add 4 each time).

Geometric Sequence

You multiply (or divide) by the same number each time. Example: 2, 6, 18, 54, … (multiply by 3).

Alternating Sequence

The pattern switches between two rules or two operations. Example: 1, 4, 2, 5, 3, 6, … (alternates +3 and −2).

Repeated Pattern

A group of numbers repeats over and over. Example: 5, 10, 15, 5, 10, 15, … (the block 5, 10, 15 repeats).

Growing Difference

The amount you add increases each step. Example: 1, 2, 4, 7, 11, … (add 1, add 2, add 3, add 4).

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KEY TAKEAWAY

Think of a number sequence like a playlist on shuffle-repeat. Once you figure out the order of the songs (the rule), you know exactly which song comes next. Your job is to listen to the first few "songs" and crack the code!

SECTION 3

Visual Explanation

The diagram below shows two common sequence types side by side. On the left you see an arithmetic sequence (add the same amount). On the right you see a geometric sequence (multiply by the same amount). Notice how the gaps between terms stay equal in the arithmetic sequence but grow larger in the geometric one.

The left side shows an arithmetic sequence where each term increases by 5. The right side shows a geometric sequence where each term is multiplied by 3. Start by checking differences, then ratios.

In the diagram, the cyan arrows on the left show a constant difference of 5 between each pair of neighbors. That tells you to add 5 to get the next term. The violet arrows on the right show a constant ratio of 3. That tells you to multiply by 3. Always try differences first. If they are all the same, you are done!

SECTION 4

Mathematical Framework

You do not need fancy formulas for the HSPT. But knowing a couple of simple equations can help you work faster. Let's look at the key formulas for the two most common sequence types.

ARITHMETIC SEQUENCE — NEXT TERM

next term = last term + d

Here, d is the common difference (the number you add each time). Find d by subtracting any term from the term right after it.

GEOMETRIC SEQUENCE — NEXT TERM

next term = last term × r

Here, r is the common ratio (the number you multiply by each time). Find r by dividing any term by the term right before it.

FINDING THE COMMON DIFFERENCE

d = term₂ − term₁

Subtract the first term from the second term. Check by also subtracting the second from the third. If they match, you have found d.

FINDING THE COMMON RATIO

r = term₂ ÷ term₁

Divide the second term by the first term. Check by also dividing the third by the second. If they match, you have found r.

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HSPT Tip

On the real test, you won't need to write out formulas. Just find the differences (or ratios) between neighbors and look for a repeating pattern. The formulas above are shortcuts for the same process.

SECTION 5

Detailed Breakdown of Sequence Types

The HSPT doesn't only test simple add-or-multiply patterns. You might also see sequences that combine operations or have changing differences. The diagram below walks through a decision process to help you identify any sequence type quickly.

This flowchart guides you through the three-step process: first check differences (arithmetic), then check ratios (geometric), and finally look for more complex patterns like alternating rules or growing differences.

Type	Example	Rule	Next Term
Arithmetic (+)	4, 9, 14, 19, …	Add 5	24
Arithmetic (−)	30, 25, 20, 15, …	Subtract 5	10
Geometric (×)	3, 12, 48, 192, …	Multiply by 4	768
Geometric (÷)	256, 64, 16, 4, …	Divide by 4	1
Growing diff.	2, 3, 5, 8, 12, …	+1, +2, +3, +4, …	17
Alternating	1, 5, 2, 6, 3, 7, …	+4, −3, +4, −3, …	4

Common sequence types you'll encounter on the HSPT

SECTION 6

Worked Example

Let's walk through a full HSPT-style problem step by step. Take your time and follow each step carefully.

ℹ

PROBLEM

What number comes next in the sequence: 5, 10, 20, 40, 80, __?

Finding the Next Term

1
Step 1 — Find the differences

Subtract each term from the next: 10 − 5 = 5, 20 − 10 = 10, 40 − 20 = 20, 80 − 40 = 40. The differences are 5, 10, 20, 40. They are not all the same, so this is NOT arithmetic.

Differences: 5, 10, 20, 40 → not constant
2
Step 2 — Try the ratios

Divide each term by the one before it: 10 ÷ 5 = 2, 20 ÷ 10 = 2, 40 ÷ 20 = 2, 80 ÷ 40 = 2. Every ratio equals 2!

Common ratio r = 2 → this is a geometric sequence
3
Step 3 — Apply the rule

Multiply the last given term by the common ratio: 80 × 2 = 160.

The next term is 160

✓

CHECK YOUR WORK

Always verify your answer by plugging it back in: 5, 10, 20, 40, 80, 160. Does 160 ÷ 80 still equal 2? Yes! You're good to go.

SECTION 7

Strategies, Strengths & Common Mistakes

Getting sequence questions right on the HSPT is all about having a clear strategy. Below is a comparison of helpful approaches versus common traps students fall into.

Good Strategy ✅	Common Mistake ❌
Write out all the differences between neighbors first.	Guessing the pattern by only looking at the first two terms.
If differences don't match, try ratios next.	Giving up when differences aren't equal instead of checking ratios.
Look at the differences of the differences (second differences).	Ignoring growing-difference patterns because they seem hard.
Check your answer by putting it back into the sequence.	Choosing an answer without verifying it fits the pattern.
Watch for alternating operations (e.g., +3 then ×2 repeating).	Assuming every sequence uses only one operation.

Strategies vs. Common Mistakes

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PATTERN DETECTIVE

Think of yourself as a detective. The numbers are your clues. Differences are like fingerprints — check them first. Ratios are like DNA evidence — try them second. And if neither works, look for a hidden two-step pattern. Follow this order every single time, and you'll crack the case!

SECTION 8

Connection to Advanced Concepts

The pattern-finding skills you are learning now will follow you through high school and beyond. In algebra, you will study sequences with formal notation. In later courses, sequences lead to powerful topics like series (adding up all the terms), functions (rules that turn an input into an output), and even calculus.

What You Learn Now	Where It Leads
Finding the common difference (d)	Slope of a linear function in Algebra 1
Finding the common ratio (r)	Exponential growth and decay in Algebra 2
Spotting growing differences	Quadratic functions (parabolas) in Algebra 1 / Geometry
Alternating patterns	Piecewise functions and series in Precalculus

How today's skills connect to future math

For now, just focus on the HSPT. But it's cool to know that every time you spot a pattern, you're building math muscles you'll use for years. The common difference d is really the same idea as slope, and the common ratio r connects to exponential growth. You are getting a head start!

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. For each one, use the difference-then-ratio strategy before you look at the answer!

Question 1: CONCEPTUAL

In the sequence 6, 12, 18, 24, …, what type of sequence is this, and what is the common difference?

Show answer

This is an arithmetic sequence. Find the common difference by subtracting neighbors: 12 − 6 = 6, 18 − 12 = 6, 24 − 18 = 6. All the differences are the same, so the common difference d = 6. You add 6 each time to get the next term.

Question 2: BASIC CALCULATION

What comes next? 7, 14, 28, 56, __

Show answer

First, check differences: 14 − 7 = 7, 28 − 14 = 14, 56 − 28 = 28. They are not the same, so try ratios: 14 ÷ 7 = 2, 28 ÷ 14 = 2, 56 ÷ 28 = 2. The common ratio r = 2. This is a geometric sequence. The next term is 56 × 2 = 112.

Question 3: INTERMEDIATE

What comes next? 1, 4, 9, 16, 25, __

Show answer

Check differences: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9. The differences are 3, 5, 7, 9 — they are not the same, but they grow by 2 each time. This is a growing-difference (quadratic) sequence. The next difference should be 9 + 2 = 11. So the next term is 25 + 11 = 36. (You may also notice these are perfect squares: 1², 2², 3², 4², 5², 6² = 36.)

Question 4: APPLIED

A bacteria colony doubles every hour. If it starts with 50 bacteria, the counts are 50, 100, 200, 400, …. How many bacteria will there be after 3 more hours?

Show answer

This is a geometric sequence with a common ratio of 2. The last given count is 400. After one more hour: 400 × 2 = 800. After two more hours: 800 × 2 = 1,600. After three more hours: 1,600 × 2 = 3,200. Each step you simply multiply by 2 because the ratio stays the same. Real-world growth often follows geometric patterns like this.

Question 5: CRITICAL THINKING

What comes next? 2, 6, 4, 12, 10, 30, __

Show answer

Check differences: 6 − 2 = 4, 4 − 6 = −2, 12 − 4 = 8, 10 − 12 = −2, 30 − 10 = 20. No constant difference. Try looking at the pattern of operations: 2 × 3 = 6, 6 − 2 = 4, 4 × 3 = 12, 12 − 2 = 10, 10 × 3 = 30. The rule alternates: multiply by 3, then subtract 2, then multiply by 3, then subtract 2, and so on. The last operation was ×3, so the next is −2. Therefore the next term is 30 − 2 = 28. Always watch for alternating operations when a single rule doesn't work!

SUMMARY

Lesson Summary

Summary

A number sequence is a list of numbers that follow a pattern. To find the pattern, start by calculating the differences between neighboring terms. If the differences are all the same, you have an arithmetic sequence with a common difference (d). If the differences vary, try dividing each term by the previous one. If those ratios are all the same, you have a geometric sequence with a common ratio (r).

When neither approach gives a single constant, look for growing differences, alternating operations, or repeating blocks. Always verify your answer by plugging the predicted next term back into the sequence to make sure it follows the same rule. Remember the three-step strategy: check differences first, then ratios, then look for fancier patterns. Master this approach and you will handle any sequence question the HSPT throws at you!