Algebraic Proofs and Pattern Recognition

This page covers several important algebraic proofs and mathematical patterns, focusing on multiples and consecutive numbers. The content demonstrates how to prove various mathematical statements through algebraic manipulation and factorization.

Definition: Algebraic proof involves using algebraic methods to demonstrate that a mathematical statement is true for all possible values within its domain.

Example: The proof that (3n+1)² - (3n-1)² is always a multiple of 12 is demonstrated through expansion and factorization: (3n+1)² - (3n-1)² = (9n² + 6n + 1) - (9n² - 6n + 1) = 12n

Highlight: The document emphasizes the importance of factorization in algebraic proofs, showing how complex expressions can be simplified to reveal patterns.

Vocabulary:

• n: represents any number
• 2n: represents any even number
• 2n+1: represents any odd number

The page includes several key proofs:

1. The sum of three consecutive even integers is always a multiple of 6
2. The sum of squares of consecutive odd numbers is always a multiple of 2
3. (2n+3)² - (2n-3)² is always a multiple of 12

Example: For consecutive even integers (2n, 2n+2, 2n+4), their sum can be written as 6n+6 or 6(n+1), proving it's always a multiple of 6.