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Pattern
Rules

Learn to identify, describe, and extend number and shape patterns by discovering the hidden rule that connects every term.

ISEE Middle Level

What Is a Pattern?

A pattern is a set of numbers, shapes, or objects arranged according to a consistent rule. Each item in the pattern is called a term. Once you find the rule, you can predict any future term without listing every one.

\( 3 \)
\( 6 \)
\( 9 \)
\( 12 \)
?
\( +3 \)
\( +3 \)
\( +3 \)
\( +3 \)

Types of Patterns

Arithmetic
Add or subtract the same number each time. Example: \( 5, 10, 15, 20, \ldots \)
Geometric
Multiply or divide by the same number each time. Example: \( 2, 6, 18, 54, \ldots \)
Repeating
A group of terms cycles over and over. Example: \( \bigcirc, \triangle, \square, \bigcirc, \triangle, \square, \ldots \)

Why Patterns Matter

Recognizing patterns lets you predict without counting every step. It also lays the groundwork for algebra: every pattern rule can be written as a formula.

Efficiency
Jump to the 100th term without listing all 99 before it.
Algebra Link
A rule like "add 4" becomes the formula \( a_n = 4n + c \).

The Core Pattern Rule

Term 1
Term 2
Term 3
Next?
rule
rule
rule
Apply the SAME rule between every pair of consecutive terms.

Arithmetic vs Geometric

Arithmetic Geometric
Operation Add or subtract Multiply or divide
Constant Common difference \( d \) Common ratio \( r \)
Example \( 4, 7, 10, 13 \) \( 3, 12, 48, 192 \)
How to find Subtract consecutive terms Divide consecutive terms
Formula \( a_n = a_1 + (n-1)d \) \( a_n = a_1 \cdot r^{\,n-1} \)

How to Find the Rule

1
List the differences. Subtract each term from the one after it: \( a_2 - a_1,\ a_3 - a_2,\ \ldots \)
2
Check for a constant. If all differences are the same, the rule is add that number.
3
If not, try ratios. Divide each term by the previous one. Equal ratios mean multiply by that number.
4
Still no match? Look for alternating operations, two-step rules (\( \times 2 + 1 \)), or repeating cycles.
5
Verify. Apply your rule to every given term. If one fails, revise.

Example 1: Add a Constant

Find the next term: \( 7, 11, 15, 19, \ldots \)

\( 7 \)
\( 11 \)
\( 15 \)
\( 19 \)
\( +4 \)
\( +4 \)
\( +4 \)
\( 11-7=4 \)
\( 15-11=4 \)
\( 19-15=4 \)

Example 2: Multiply by a Constant

Find the next term: \( 5, 15, 45, 135, \ldots \)

\( 5 \)
\( 15 \)
\( 45 \)
\( 135 \)
\( \times 3 \)
\( \times 3 \)
\( \times 3 \)
\( 15\div 5=3 \)
\( 45\div 15=3 \)
\( 135\div 45=3 \)

Example 3: Two-Step Rule

Find the next term: \( 1, 3, 7, 15, 31, \ldots \)

1
Try differences: \( 2, 4, 8, 16 \). Not constant, but they double!
2
Try ratios: \( 3/1 = 3,\ 7/3 \approx 2.3 \). Not constant either.
3
Look deeper: Each term is \( \times 2 + 1 \). Check: \( 1 \times 2 + 1 = 3 \), \( 3 \times 2 + 1 = 7 \), \( 7 \times 2 + 1 = 15 \). ✓

Visual Shape Patterns

Patterns can also appear as shapes. Count the objects in each figure to find the number pattern underneath.

Fig 1
\( 1 \)
Fig 2
\( 3 \)
Fig 3
\( 5 \)
Fig 4
\( 7 \)

Writing a Formula

Once you know the rule, you can write a formula to find any term directly. For the pattern \( 7, 11, 15, 19, \ldots \) with \( d = 4 \) and first term \( a_1 = 7 \):

✓
Test \( n = 1 \): \( 4(1) + 3 = 7 \) ✓
✓
Test \( n = 4 \): \( 4(4) + 3 = 19 \) ✓
✓
Find \( n = 50 \): \( 4(50) + 3 = 203 \). Done instantly!

Decreasing Patterns

Not every pattern grows. Some decrease by subtracting or dividing.

\( 100 \)
\( 85 \)
\( 70 \)
\( 55 \)
\( −15 \)
\( −15 \)
\( −15 \)

Common Mistakes

✕ Checking Only One Pair
A rule must work for every consecutive pair, not just the first two terms.
✕ Confusing Add vs Multiply
Differences of \( 6, 6, 6 \) means add \( 6 \). Ratios of \( 3, 3, 3 \) means multiply by \( 3 \). Don't mix them up.
✕ Forgetting Negative Rules
A decreasing pattern uses a negative difference. Don't assume every pattern increases.

Summary of Pattern Types

Pattern Type How to Detect Example
Add constant Same difference \( 2, 5, 8, 11 \)
Subtract constant Same negative diff. \( 50, 42, 34, 26 \)
Multiply constant Same ratio \( 4, 12, 36, 108 \)
Divide constant Same fraction ratio \( 81, 27, 9, 3 \)
Two-step rule Changing diffs with a pattern \( 1, 3, 7, 15 \)
Repeating cycle Terms repeat in a loop \( A, B, C, A, B, C \)

Question 1 of 5

What is the next term in the pattern \( 4, 10, 16, 22, \ldots \)?

Question 2 of 5

Which rule describes the pattern \( 3, 9, 27, 81, \ldots \)?

Question 3 of 5

The pattern \( 50, 43, 36, 29, \ldots \) follows the rule "subtract \( 7 \)." What is the 6th term?

Question 4 of 5

A pattern starts at \( 2 \) and uses the rule "multiply by \( 2 \) then add \( 1 \)." What is the 4th term?

Question 5 of 5

In the repeating pattern A, B, C, D, A, B, C, D, ..., what is the 23rd term?