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Recognizing
Patterns

Learn to spot repeated reasoning, discover shortcuts and generalizations, and explain the patterns hiding inside numbers, shapes, and expressions.

Pre-Algebra

What Is a Pattern?

A pattern is a rule that tells you how a sequence of numbers, shapes, or events behaves. Once you find the rule, you can predict what comes next and generalize the idea to new situations.

Sequence:
\(2\)
\(4\)
\(6\)
\(8\)
?
\(+2\)
\(+2\)
\(+2\)
\(+2\)

Why Patterns Matter

★ Predict
Once you know the rule, you can find the \(100\)th term without listing all \(99\) before it.
★ Simplify
Patterns let you replace slow, repetitive work with a single formula or shortcut.
★ Generalize
A pattern that works for specific numbers can be written as a rule that works for all numbers.

Three Types of Patterns

Arithmetic
Add (or subtract) the same amount each step. Example: \(3, 7, 11, 15, \ldots\) adds \(4\) each time.
Geometric
Multiply (or divide) by the same amount each step. Example: \(2, 6, 18, 54, \ldots\) multiplies by \(3\).
Other / Mixed
Squares, triangular numbers, Fibonacci, or custom rules. Example: \(1, 4, 9, 16, \ldots\) are perfect squares.

The Arithmetic Pattern Rule

An arithmetic pattern has a constant difference \(d\) between consecutive terms. If the first term is \(a_1\), then:

\(5\)
\(n=1\)
\(8\)
\(n=2\)
\(11\)
\(n=3\)
\(14\)
\(n=4\)
\(+3\)
\(+3\)
\(+3\)
\(a_1 = 5,\; d = 3\) so \(a_n = 5 + 3(n-1) = 3n + 2\)

The Geometric Pattern Rule

A geometric pattern has a constant ratio \(r\) between consecutive terms. If the first term is \(a_1\), then:

\(3\)
\(n=1\)
\(6\)
\(n=2\)
\(12\)
\(n=3\)
\(24\)
\(n=4\)
\(\times 2\)
\(\times 2\)
\(\times 2\)
\(a_1 = 3,\; r = 2\) so \(a_n = 3 \cdot 2^{n-1}\)

Arithmetic vs Geometric

Arithmetic Geometric
Operation Add / Subtract Multiply / Divide
Constant Common difference \(d\) Common ratio \(r\)
Formula \(a_1 + (n-1)d\) \(a_1 \cdot r^{n-1}\)
Growth Linear (steady) Exponential (rapid)
Example \(5, 8, 11, 14, \ldots\) \(3, 6, 12, 24, \ldots\)

How to Find a Pattern

1
List the terms. Write them out clearly so you can compare.
2
Check differences. Subtract each term from the next: are the differences the same?
3
Check ratios. Divide each term by the previous one: are the ratios the same?
4
Look deeper. If neither works, check second differences, squares, cubes, or other operations.
5
Write the rule. Express the pattern as a formula using \(n\), then verify it with known terms.

Example 1: Arithmetic Sequence

Find the \(20\)th term of \(7, 12, 17, 22, \ldots\)

\(7\)
\(12\)
\(17\)
\(22\)
\(+5\)
\(+5\)
\(+5\)
\(a_1 = 7, \quad d = 5\)

Example 2: Geometric Sequence

Find the \(6\)th term of \(4, 12, 36, 108, \ldots\)

\(4\)
\(12\)
\(36\)
\(108\)
\(\times 3\)
\(\times 3\)
\(\times 3\)
\(a_1 = 4, \quad r = 3\)

Example 3: Square Number Pattern

What is the pattern in \(1, 4, 9, 16, 25, \ldots\)?

\(1^2\)
\(2^2\)
\(3^2\)
\(4^2\)
\(+3\)
\(+5\)
\(+7\)
Differences grow by \(+2\) each time (odd numbers!)
Rule: the \(n\)th term is \(n^2\)

Visualizing Linear vs Exponential

\(n\) (term number)
value
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(0\)
\(5\)
\(10\)
\(15\)
\(20\)
Arithmetic
Geometric

Using Second Differences

When first differences are not constant, compute the differences of the differences (second differences). If those are constant, the rule involves \(n^2\).

Terms:
\(2\)
\(5\)
\(10\)
\(17\)
\(26\)
1st diff:
\(3\)
\(5\)
\(7\)
\(9\)
2nd diff:
\(2\)
\(2\)
\(2\)
Constant second difference of \(2\) → quadratic pattern!
Rule: \(a_n = n^2 + 1\)

From Pattern to Formula

The real power of recognizing a pattern is turning it into a formula: a generalization that works for any term number \(n\).

1
Identify the type (arithmetic, geometric, quadratic, etc.).
2
Find key values: first term \(a_1\), common difference \(d\), ratio \(r\), or second difference.
3
Plug into the formula and simplify.
4
Verify by testing the first \(2\)-\(3\) known terms.

Common Mistakes

✕ Off-by-one Error
Using \(n\) instead of \((n-1)\) in the formula. The formula \(a_n = a_1 + nd\) gives the wrong answer because it counts one extra step.
✕ Confusing Types
Treating a geometric pattern as arithmetic (adding instead of multiplying). Always check ratios if differences aren't constant.
✕ Skipping Verification
Writing a formula but never checking it against known terms. A quick plug-in test catches most errors instantly.

Pattern Summary

Pattern Type Key Test Formula Example
Arithmetic Constant 1st differences \(a_1 + (n-1)d\) \(5, 8, 11, 14\)
Geometric Constant ratios \(a_1 \cdot r^{n-1}\) \(3, 6, 12, 24\)
Quadratic Constant 2nd differences \(an^2 + bn + c\) \(1, 4, 9, 16\)
Other Look for known sequences Varies \(1, 1, 2, 3, 5, 8\)

Question 1 of 5

The sequence \(6, 11, 16, 21, 26, \ldots\) is arithmetic. What is the \(50\)th term?

Question 2 of 5

What is the common ratio of the geometric sequence \(5, 15, 45, 135, \ldots\)?

Question 3 of 5

The sequence \(2, 5, 10, 17, 26, \ldots\) has constant second differences. What type of pattern is this?

Question 4 of 5

A student writes the formula \(a_n = 4 + 3n\) for the sequence \(4, 7, 10, 13, \ldots\). Is this correct?

Question 5 of 5

What is the \(8\)th term of the geometric sequence \(2, 6, 18, 54, \ldots\)?