Uncovering Number Patterns: Sequences

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• Mathematics methods
• Secondary Education

Uncovering Number Patterns: Sequences

What do you already know about number patterns?

Learning Objective

At the end of the lesson you will identify, describe, and continue number patterns, and classify them as types of sequences.

What Are Number Patterns?

A number pattern is a sequence of numbers that follow a certain rule or relationship.

Identifying Patterns

Look for changes between numbers. Is the sequence adding, subtracting, multiplying, or dividing?

Example: Simple Addition Pattern

Sequence: 2, 4, 6, 8, ... Rule: Add 2 each time.

Challenge: Find the Rule

What is the rule for: 5, 10, 15, 20, ...?

Arithmetic Sequences

An arithmetic sequence has a constant difference between terms.

Example: Arithmetic Sequence

3, 7, 11, 15, ... Difference = 4. The rule: Add 4 each time.

Describing Arithmetic Sequences

Example

Sequence: 5, 8, 11, 14, ... nth term = 5 + (n-1) × 3

nth term = first term + (n-1) × difference.

Formula

Practice: Find the 10th Term

In the sequence: 1, 4, 7, 10, ... what is the 10th term?

Geometric Sequences

A geometric sequence has a constant ratio between terms.

Example: Geometric Sequence

2, 4, 8, 16, ... Rule: Multiply by 2 each time.

Describing Geometric Sequences

Sequence: 3, 6, 12, 24, ... nth term = 3 × 2^(n-1)

Formula

nth term = first term × (ratio)^(n-1).

Example

Practice: Geometric Pattern

Find the 6th term in: 5, 15, 45, 135, ...

Recognizing Other Patterns

Not all sequences are arithmetic or geometric. Some use other rules, such as squares or primes.

Example: Prime Number Pattern

2, 3, 5, 7, 11, ... (prime numbers in order)

Exploring Fibonacci Sequence

Each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, ...

Real-World Uses of Sequences

Patterns appear in nature, music, engineering, and finance. Sequences help us make predictions.

Nature’s Patterns

Fibonacci numbers are found in flower petals, pinecones, and shells.

Review: Key Ideas

Multiply or divide by the same number.

Arithmetic

Geometric

Add or subtract by the same number.

What’s Next?

Practice finding sequences in the world around you, and try creating your own number patterns!

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