Understanding Patterns and Sequences

Concept Explanation

Patterns are regular arrangements of numbers, shapes, or other elements that follow a rule. Recognizing patterns helps us predict what comes next and is a fundamental skill in algebraic thinking. Patterns can be repeating (like ABAB) or growing (like 2, 4, 6, 8…).

A sequence is an ordered list of numbers that follow a pattern or rule. Finding the rule allows us to extend the sequence and find any term in it.

Worked Examples

Example 1

Problem: Find the next two numbers in the sequence: 5, 10, 15, 20, …
Solution: 25, 30
Explanation: Each number increases by 5, so after 20 comes 25, then 30.

Example 2

Problem: Describe the pattern: 3, 6, 12, 24, …
Solution: Each number is multiplied by 2
Explanation: 3 × 2 = 6, 6 × 2 = 12, 12 × 2 = 24, so the pattern is “multiply by 2.”

Example 3

Problem: Find the missing number: 2, 6, 18, ___, 162
Solution: 54
Explanation: Each number is multiplied by 3, so 18 × 3 = 54.

Common Errors

Practice Problems

1. Problem: What comes next: 7, 14, 21, 28, …?
   Solution: 35
2. Problem: Find the pattern: 1, 4, 9, 16, 25, …
   Solution: Square numbers: 1², 2², 3², 4², 5², …
3. Problem: Continue the sequence: 80, 40, 20, 10, …
   Solution: 5, 2.5
4. Problem: What’s the missing number: 3, 7, 15, 31, ___
   Solution: 63
5. Problem: Describe the pattern: 1, 3, 6, 10, 15, …
   Solution: Triangle numbers: add 1, then 2, then 3, etc.

Real-World Application Example

Patterns appear everywhere in our world—from the arrangement of leaves on a stem to the rhythm of music. Understanding patterns helps scientists predict weather, economists forecast market trends, and engineers design efficient structures. This skill is essential for problem-solving and making predictions based on existing information.