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Pattern Rules
From Sequences

Learn to identify the hidden rule that generates a sequence of numbers and predict what comes next.

SSAT Middle Level: Quantitative

What Is a Sequence?

A sequence is an ordered list of numbers that follows a specific rule. Each number in the list is called a term. The position of each term matters: the first term, second term, third term, and so on.

3 1st 6 2nd 9 3rd 12 4th ? 5th
\(+3\)
\(+3\)
\(+3\)
\(+3\)

What Is a Pattern Rule?

A pattern rule is the instruction that tells you how to get from one term to the next, or how to calculate any term directly from its position number. Finding the pattern rule lets you predict any term in the sequence without listing them all.

Recursive Rule
Describes how to get the next term from the current term. Example: "add 5 each time."
Position Rule
Gives a formula using the term's position number \( n \). Example: the \( n \)th term is \( 5n \).

Why Do Pattern Rules Matter?

On the SSAT, you will see sequences and be asked to find the next term, a missing term, or the rule itself. Spotting the pattern quickly saves time and avoids errors.

Predict
Find the 10th or 100th term without writing every single term.
Verify
Check if a number belongs to a given sequence by testing the rule.
Generalize
Write a formula that works for any position in the sequence.

Common Types of Patterns

Type Operation Example Rule
Arithmetic (Add) \( \text{+ constant} \) 2, 5, 8, 11, ... Add 3
Arithmetic (Subtract) \( \text{− constant} \) 20, 17, 14, 11, ... Subtract 3
Geometric (Multiply) \( \times \text{ constant} \) 3, 6, 12, 24, ... Multiply by 2
Geometric (Divide) \( \div \text{ constant} \) 81, 27, 9, 3, ... Divide by 3
Two-Step \( \times \text{ then } + \) 1, 3, 7, 15, ... \( \times 2 + 1 \)

Arithmetic vs Geometric

Arithmetic Sequence
The difference between consecutive terms is constant. You add or subtract the same number each time.
Geometric Sequence
The ratio between consecutive terms is constant. You multiply or divide by the same number each time.

The Position Formula

For an arithmetic sequence, you can write a formula for the \( n \)th term:

4 n = 1 9 n = 2 14 n = 3 19 n = 4 24 n = 5
\(+5\)
\(+5\)
\(+5\)
\(+5\)
\( a_1 = 4, d = 5 \rightarrow a_n = 4 + (n-1) \times 5 = 5n - 1 \)

How to Find the Pattern Rule

1
List the differences. Subtract each term from the next: \( a_2 - a_1 \), \( a_3 - a_2 \), and so on.
2
Check for a constant difference. If all differences are the same, you have an arithmetic pattern.
3
If not, try ratios. Divide each term by the previous one. If all ratios are equal, it is geometric.
4
If neither works, look for two-step rules. Try combining operations like "multiply then add" or "square plus constant."
5
Verify your rule. Apply it to every term in the sequence. If it works for all, you found the pattern.

Example 1: Arithmetic Sequence

Find the pattern rule and the next term: 7, 11, 15, 19, ...

7 11 15 19 23
\(+4\)
\(+4\)
\(+4\)
\(+4\)
 1st 2nd 3rd 4th 5th

Example 2: Geometric Sequence

Find the pattern rule and the next term: 2, 6, 18, 54, ...

2 6 18 54 162
\(\times 3\)
\(\times 3\)
\(\times 3\)
\(\times 3\)
 1st 2nd 3rd 4th 5th

Example 3: Two-Step Rule

Find the pattern rule and the next term: 1, 3, 7, 15, 31, ...

1 3 7 15 31
\(+2\)
\(+4\)
\(+8\)
\(+16\)
 1st 2nd 3rd 4th 5th
Differences are doubling: 2, 4, 8, 16 → Rule: \( \times 2 + 1 \)

Example 4: Decreasing Pattern

Find the pattern rule and the next term: 100, 93, 86, 79, ...

100 93 86 79 72
\(-7\)
\(-7\)
\(-7\)
\(-7\)
 1st 2nd 3rd 4th 5th

Example 5: Position-Based Pattern

Find the rule: 1, 4, 9, 16, 25, ...

1 4 9 16 25 n = 1 n = 2 n = 3 n = 4 n = 5
\( 1^2 \)
\( 2^2 \)
\( 3^2 \)
\( 4^2 \)
\( 5^2 \)
\( \text{Each term} = (\text{position number})^2 \)

Common Mistakes to Avoid

✕ Checking Only Two Terms
A rule that works for the first two terms might not work for all. Always verify with at least three consecutive differences or ratios.
✕ Confusing Add with Multiply
2, 4, 8, 16 uses \( \times 2 \), not \( +2 \). If differences are growing, check for multiplication before addition.
✕ Ignoring Negative Differences
A decreasing sequence has a negative common difference. Don't forget the minus sign when writing the rule.

Summary: Pattern Rule Toolkit

Step What to Do Result
1. Differences Subtract consecutive terms Constant → Arithmetic
2. Ratios Divide consecutive terms Constant → Geometric
3. Position link Compare term to \( n \), \( n^2 \), etc. Direct formula found
4. Two-step Combine operations (\( \times \), then \( + \)) Recursive rule found
5. Verify Test rule on every given term Confidence in answer

Question 1 of 5

What is the next term in the sequence 5, 12, 19, 26, ... ?

Question 2 of 5

Which rule describes the sequence 3, 9, 27, 81, ... ?

Question 3 of 5

In the sequence 50, 43, 36, 29, ..., what is the 6th term?

Question 4 of 5

The sequence 1, 4, 9, 16, 25, ... follows the rule \( a_n = n^2 \). What is the 8th term?

Question 5 of 5

What is the next term in the sequence 2, 5, 11, 23, 47, ... ?