Identify a pattern rule from a sequence. - SSAT Middle Level Quantitative

SECTION 1

Why Do Patterns Matter?

People have been fascinated by number patterns for thousands of years. A sequence (an ordered list of numbers that follow a rule) is one of the oldest ideas in math. Ancient civilizations used sequences to track the seasons, predict eclipses, and build some of the most amazing structures on Earth.

~2000 BCE • Babylonian Number Tables

Babylonian scribes carved tables of squares (1, 4, 9, 16, 25 …) and cubes into clay tablets. These helped them solve early algebra problems.

~300 BCE • Euclid and the Greeks

Greek mathematicians studied figurate numbers — sequences made by arranging dots into triangles or squares. Euclid also explored prime numbers as a special sequence.

1202 • Fibonacci's Famous Sequence

Leonardo Fibonacci introduced a sequence (1, 1, 2, 3, 5, 8 …) where each number is the sum of the two before it. This pattern appears in sunflowers, seashells, and even galaxies.

1700s • Arithmetic & Geometric Series

Mathematicians like Euler and Gauss developed formulas for adding up long sequences quickly. The young Gauss famously added the numbers 1 through 100 in seconds!

Today • Patterns Everywhere

Modern computers search for patterns in weather data, stock prices, DNA, and more. Recognizing a pattern rule is the first step in making predictions.

At its heart, finding a pattern rule is about asking a simple question: "What is happening from one number to the next?" Once you crack that rule, you can predict any number in the sequence — even the 100th or the 1,000th!

SECTION 2

Core Ideas Behind Pattern Rules

Before you can find a rule, you need to understand a few key ideas. These four building blocks will help you tackle any sequence you see on the SSAT.

Term

Each number in a sequence is called a term. In the sequence 3, 7, 11, 15, the first term is 3, the second term is 7, and so on.

Common Difference

When you add (or subtract) the same number each time, that number is the common difference. In 3, 7, 11, 15, you add 4 every step.

Common Ratio

When you multiply (or divide) by the same number each time, that number is the common ratio. In 2, 6, 18, 54, you multiply by 3 every step.

Position Number

The position number tells you where a term sits. The 1st position holds term 1, the 2nd holds term 2, etc. Rules often connect the position number to the term's value.

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KEY TAKEAWAY

Think of a sequence like a vending machine. You press a button (the position number) and out pops a snack (the term). The pattern rule is the hidden wiring inside the machine that decides which snack matches which button. Your job is to figure out how the machine works by looking at what comes out!

SECTION 3

Seeing the Pattern

One of the best ways to spot a rule is to draw out the differences between terms. The diagram below shows two common types of sequences and how the "jumps" between terms reveal the rule.

In the top row, every curved arrow shows the same jump of +3 (arithmetic). In the bottom row, every arrow shows a jump of ×3 (geometric). Always check the jumps first!

The trick is simple. Start by finding the difference between the first term and the second term. Then check if that same difference appears between the second and third, the third and fourth, and so on. If it does, you have an arithmetic sequence. If the differences keep changing, try dividing each term by the one before it. If the answer is always the same, you have a geometric sequence.

SECTION 4

Formulas for Common Sequences

You do not need to memorize complicated formulas for the SSAT. But knowing these two simple patterns will help you work faster.

ARITHMETIC RULE

term = first term + (position − 1) × common difference

first term = the starting number. position = which term you want (1st, 2nd, 3rd …). common difference = the number added each step.

For example, if a sequence starts at 5 and adds 3 each time, the 10th term is 5 + (10 − 1) × 3 = 5 + 27 = 32.

GEOMETRIC RULE

term = first term × ratio^(position − 1)

ratio = the number you multiply by each step. The little ^ means "raised to the power of."

For example, if a sequence starts at 2 and multiplies by 3 each time, the 5th term is 2 × 3⁴ = 2 × 81 = 162.

OTHER PATTERNS

Look for squares (n²), cubes (n³), or alternating operations.

Not every sequence is arithmetic or geometric. Some use perfect squares (1, 4, 9, 16, 25 …) or mix operations like "+2, then ×3, then +2, then ×3." Always check whether the differences change in a pattern of their own.

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SSAT Tip

On the SSAT, you usually only need to find the next one or two terms. You rarely need to jump to the 50th or 100th term. So focus on figuring out the step-by-step rule first, then apply it once or twice.

SECTION 5

Types of Pattern Rules You Will See

The SSAT tests several kinds of sequences. The diagram below organizes them so you can quickly decide which type you are dealing with.

Follow the flowchart from top to bottom. Start by subtracting consecutive terms. If the differences are all the same, it is arithmetic. If not, try dividing. Same quotient every time means geometric. Otherwise, look for square numbers or alternating rules.

Type	Example	Rule	Next Term
Arithmetic (add)	4, 9, 14, 19, …	Add 5	24
Arithmetic (subtract)	30, 24, 18, 12, …	Subtract 6	6
Geometric (multiply)	3, 12, 48, 192, …	Multiply by 4	768
Geometric (divide)	256, 64, 16, 4, …	Divide by 4	1
Perfect squares	1, 4, 9, 16, 25, …	n²	36
Alternating	2, 6, 4, 8, 6, …	+4, −2, +4, −2	10

Common sequence types on the SSAT

SECTION 6

Worked Example: Finding the Rule Step by Step

Let's walk through a complete problem, just like one you might see on the SSAT.

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PROBLEM

What is the next number in this sequence? 7, 11, 15, 19, 23, ___

Solution

1
Step 1 — Find the Differences

Subtract each term from the one that follows it. 11 − 7 = 4. 15 − 11 = 4. 19 − 15 = 4. 23 − 19 = 4.

Every difference is 4.
2
Step 2 — Name the Pattern Type

Because the difference is the same every time, this is an arithmetic sequence with a common difference of 4.
3
Step 3 — State the Rule

The rule is: "Add 4 to the previous term to get the next term."
4
Step 4 — Apply the Rule

The last given term is 23. Adding 4 gives us: 23 + 4 = 27.

The next term is 27.
5
Step 5 — Quick Check

Double-check by making sure 27 − 23 = 4. It does! The pattern stays the same, so we are confident in our answer.

✓

ALWAYS VERIFY

After you find the rule, apply it once and check. This is like spell-checking an essay — it takes only a few seconds and can save you from picking the wrong answer.

SECTION 7

Strategies and Common Mistakes

Knowing the types of sequences is great, but the SSAT can also test your problem-solving skills with tricky answer choices. Here are strategies that work, along with mistakes to avoid.

Strategy ✅	Common Mistake ❌
Write out ALL differences between terms before deciding the rule.	Checking only the first pair and assuming the same rule continues.
If differences are not equal, try dividing consecutive terms next.	Giving up and guessing when subtraction does not reveal a pattern.
Look at the differences of the differences (second differences) for square-number patterns.	Ignoring second differences, which causes you to miss quadratic patterns.
Watch for alternating signs or operations (+, −, +, −).	Treating every sequence as if it uses one operation.
Double-check your answer by plugging it back in.	Rushing to the next question without verifying.

Do's and Don'ts for sequence problems

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THINK LIKE A DETECTIVE

Finding a pattern rule is like solving a mystery. The terms are your clues, the differences are fingerprints, and the rule is the answer that ties everything together. Don't jump to conclusions after one clue — gather all the evidence first!

SECTION 8

From Patterns to Algebra

Pattern rules are actually the starting point for a bigger idea you will meet in algebra: functions. A function is just a rule that takes an input (like a position number) and gives you an output (the term). The table below shows how what you already know connects to what comes next.

What You Know Now	What It Becomes in Algebra
"Add 3 each time, starting at 5"	y = 3x + 2 (a linear function)
"Multiply by 2 each time, starting at 3"	y = 3 × $2^{x−1}$ (an exponential function)
"The term equals the position squared"	y = x² (a quadratic function)
Common difference	Slope of a line
Position number	The variable x

Pattern rules lead directly to algebra concepts

You don't need to know these algebra terms for the SSAT right now. The important thing is that every time you find a pattern rule, you are building the exact skill that will make algebra easier later. Think of it as training for a bigger race!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In the sequence 10, 15, 20, 25, 30, …, which best describes the pattern rule? (A) Multiply by 5 (B) Add 5 (C) Subtract 5 (D) Multiply by 2 (E) Add 10

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(B) Add 5. Find the differences: 15 − 10 = 5, 20 − 15 = 5, 25 − 20 = 5, 30 − 25 = 5. Every difference is 5, so the rule is "add 5." Choice (A) fails because 10 × 5 = 50, not 15. Choice (C) gives decreasing numbers. Choice (D) gives 10, 20, 40 — wrong. Choice (E) gives 10, 20, 30 — wrong second term.

Question 2: BASIC CALCULATION

What is the next number in the sequence 3, 6, 12, 24, ___? (A) 30 (B) 36 (C) 48 (D) 72 (E) 96

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(C) 48. Check the differences: 6 − 3 = 3, 12 − 6 = 6, 24 − 12 = 12. The differences are not the same, so try dividing: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. This is a geometric sequence with a common ratio of 2. Multiply 24 × 2 = 48. Choice (A) adds 6 (wrong operation). Choice (B) adds 12. Choice (D) multiplies by 3. Choice (E) multiplies by 4.

Question 3: INTERMEDIATE

Find the missing number: 2, 5, 10, 17, ___, 37. (A) 22 (B) 24 (C) 26 (D) 28 (E) 30

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(C) 26. Find the differences: 5 − 2 = 3, 10 − 5 = 5, 17 − 10 = 7. The differences themselves are increasing by 2 each time (3, 5, 7, …). The next difference should be 9, so 17 + 9 = 26. Check: 37 − 26 = 11, and 11 is indeed 9 + 2, confirming the pattern. Choice (A) uses a difference of 5. Choice (B) uses 7. Choice (D) uses 11. Choice (E) uses 13 — all break the pattern of increasing by 2.

Question 4: APPLIED

A movie theater has 12 seats in the first row, 16 seats in the second row, and 20 seats in the third row. If this pattern continues, how many seats are in the sixth row? (A) 24 (B) 28 (C) 30 (D) 32 (E) 36

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(D) 32. The differences are 16 − 12 = 4 and 20 − 16 = 4, so this is an arithmetic sequence with a common difference of 4. Continue: Row 4 = 20 + 4 = 24, Row 5 = 24 + 4 = 28, Row 6 = 28 + 4 = 32. Alternatively, use the formula: term = 12 + (6 − 1) × 4 = 12 + 20 = 32. Choice (A) is only the 4th row. Choice (B) is the 5th row. Choice (C) uses a common difference of 3.6. Choice (E) multiplies 12 by 3, which is not the rule.

Question 5: CRITICAL THINKING

Look at this sequence: 1, 2, 4, 7, 11, 16, ___. What is the next number? (A) 18 (B) 20 (C) 22 (D) 23 (E) 25

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(C) 22. This is neither arithmetic nor geometric. Find the first differences: 2 − 1 = 1, 4 − 2 = 2, 7 − 4 = 3, 11 − 7 = 4, 16 − 11 = 5. The differences are 1, 2, 3, 4, 5 — they increase by 1 each time. So the next difference should be 6, giving 16 + 6 = 22. Choice (A) adds only 2. Choice (B) adds 4. Choice (D) adds 7 (confusing first differences with terms). Choice (E) adds 9. The key insight is examining the second-level pattern within the differences.

SUMMARY

Putting It All Together

Summary

A sequence is an ordered list of numbers that follows a rule. To find the pattern rule, start by finding the differences between consecutive terms. If every difference is the same, you have an arithmetic sequence — the rule is to add (or subtract) that common difference. If the differences are not equal, try dividing consecutive terms. A constant quotient means a geometric sequence with a common ratio.

For trickier sequences, look at the differences of the differences to spot patterns involving perfect squares or increasing gaps. Watch for alternating patterns that switch between two operations. Always verify your answer by checking that it fits the rule. Use the position number and the formulas when you need to jump ahead in a sequence. Mastering these skills now will prepare you for algebraic functions later on!