1) What is the main purpose of a proof in mathematics? a) To calculate numerical results b) To convince others using authority c) To logically demonstrate that a statement is true d) To provide a counterexample 2) Which type of proof begins by assuming the negation of the statement and arriving at a contradiction? a) Direct Proof b) Proof by Induction c) Proof by Contradiction d) Proof by Counterexample 3) Which of the following is an example of a direct proof? a) Proving √2 is irrational by assuming it is rational b) Showing that if n is even, then n² is even c) Using induction to prove a formula for ∑n d) Proving there are infinitely many primes using contradiction 4) What is the first step in a mathematical induction proof? a) Assume the statement is true for all integers b) Show the statement is true for the base case c) Assume the statement is false d) Provide a counterexample 5) In proof by induction, what do you do after proving the base case? a) Prove the statement for infinitely many cases b) Assume the statement is true for k and prove it for k+1 c) Provide a geometric diagram d) Use a counterexample 6) Which of the following best describes a counterexample? a) An example that illustrates the statement is true b) A specific example that shows the statement is false c) A method of proof that works for all integers d) A type of direct proof 7) What type of proof is most commonly used to establish divisibility properties in number theory? a) Proof by Contradiction b) Proof by Induction c) Geometric Proof d) Proof by Counterexample 8) Which of the following statements requires a proof by contradiction to establish? a) If a number is divisible by 6, it is divisible by 2 b) The sum of two even numbers is even c) √3 is irrational d) The sum of the first n natural numbers is (n(n+1))/2 9) Which of the following is an example of a statement that can be proved by induction? a) The angles in a triangle add to 180° b) There are infinitely many prime numbers c) For all n ≥ 1, 1 + 2 + … + n = n(n+1)/2 d) √2 is irrational 10) In a proof, why is it not enough to verify a statement with several numerical examples? a) Because numerical examples cannot exist in mathematics b) Because examples may only show the pattern, not guarantee it for all cases c) Because checking numbers is too time-consuming d) Because only counterexamples are valid

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