Proofs

Which of the following is a valid method of proof to show that \( \sqrt{2} \) is irrational?

If \( n \) is an integer and \( n^2 \) is even, which of the following statements is true?

Which proof technique is most appropriate to prove the formula for the sum of the first \( n \) natural numbers, \( S = \frac{n(n+1)}{2} \)?

In proving the statement "For all integers \( n \), \( n^3 - n \) is divisible by 6," which of the following is a key step?

Which of the following statements is true about proof by contradiction?

Consider the statement: "For all integers \( n \geq 1 \), \( 7^n - 1 \) is divisible by 6." Which method is best suited to prove this?

Which of the following is NOT a valid step in a proof by induction?

Given the sequence defined by \( a_1 = 2 \) and \( a_{n+1} = 3a_n + 4 \), which method would you use to prove the closed form formula for \( a_n \)?

Which of the following is a correct way to prove that the function \( f(x) = x^2 \) is increasing for \( x \geq 0 \)?

In the context of IB Mathematics Analysis and Approaches, which of the following best describes the role of axioms in proofs?