Algebraic Proof Flashcards

A logical argument using algebra to show that a statement is always true.

An even number can be written as 2n, where n is an integer.

An odd number can be written as 2n + 1, where n is an integer.

The sum of two even numbers is always even. Proof: (2n) + (2m) = 2(n + m), which is divisible by 2.

The sum of two odd numbers is always even. Proof: (2n + 1) + (2m + 1) = 2(n + m + 1), which is divisible by 2.

The product of two odd numbers is always odd. Proof: (2n + 1)(2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1.

a² - b² = (a + b)(a - b). This is a key identity used in algebraic proofs.

If a number can be written as 3n, where n is an integer, it is divisible by 3.

(2n + 1)² = 4n² + 4n + 1 = 2(2n² + 2n) + 1, which is odd.

(n) + (n + 1) = 2n + 1, which is odd for any integer n.