1) What is the base case in mathematical induction? a) A) The inductive step b) B) The smallest value for which the statement is true c) C) The largest value for which the statement is true d) D) The statement to be proven 2) What is the inductive step in mathematical induction? a) A) Assuming the statement is true for a specific value b) B) Proving the statement is true for all values c) C) Assuming the statement is true for k and proving it's true for k+1 d) D) Proving the statement is true for the base case 3) Which of the following is a common technique used in mathematical induction? a) A) Direct proof b) B) Proof by contradiction c) C) Strong induction d) D) All of the above 4) What is the difference between weak induction and strong induction? a) A) Weak induction assumes the statement is true for k, while strong induction assumes it's true for all values less than or equal to k. b) B) Strong induction assumes the statement is true for k, while weak induction assumes it's true for all values less than or equal to k. c) C) Weak induction is used for simpler proofs, while strong induction is used for more complex proofs. d) D) There is no difference between weak and strong induction. 5) Can mathematical induction be used to prove statements about real numbers? a) A) True b) B) False c) C) Only for certain types of statements d) D) It depends on the specific statement 6) Use mathematical induction to prove that 1 + 2 + 3 + ... + n = n(n+1)/2 for all positive integers n. a) A) The statement is true for all n. b) B) The statement is only true for n = 1. c) C) The statement is not true for any n. d) D) The statement is true for all n except for n = 1. 7) Prove that 2^n > n for all positive integers n greater than or equal to 3. a) A)The statement is true for all n except for n = 3. b) B) The statement is only true for n = 3. c) C) The statement is not true for any n. d) D)The Statement is True for all N 8) Use mathematical induction to prove that n^3 + 5n is divisible by 3 for all positive integers n. a) A) The statement is true for all n. b) B) The statement is only true for n = 1. c) C) The statement is not true for any n. d) D) The statement is true for all n except for n = 1. 9) Prove that 1 + 3 + 5 + ... + (2n-1) = n^2 for all positive integers n. a) A) The statement is only true for n = 1. is true for all n. b) B) The statement is true for all n. c) C) The statement is not true for any n. d) D) The statement is true for all n except for n = 1. 10) What is the principle of strong induction? a) A) Assuming the statement is true for k and proving it's true for k+1 b) B) Assuming the statement is true for all values less than or equal to k and proving it's true for k+1 c) C) Assuming the statement is true for k and proving it's true for k-1 d) D) Assuming the statement is true for all values greater than or equal to k and proving it's true for k-1 11) Why is the inductive step crucial in mathematical induction? a) A) It proves the statement for the base case. b) B) It shows that the statement is true for all values of n. c) C) It demonstrates that if the statement is true for a particular value, it must also be true for the next value. d) D) It is used to determine the base case. 12) Prove that n^2 + n is even for all positive integers n. a) A) The statement is true for all n. b) B) The statement is only true for n = 1. c) C) The statement is not true for any n. d) D) The statement is true for all n except for n = 1. 13) Prove that n^5 - n is divisible by 5 for all positive integers n. a) A) The statement is unachievable b) B) The statement is only true for n = 1. c) C) The statement is true for all n d) D) The statement is true for all n except for n = 1. 14) Prove that n! > 2^n for all positive integers n greater than or equal to 4. a) A) The statement is unachievable b) B) The statement is only true for n = 4. c) C) The statement is true for all n d) D) The statement is true for all n except for n = 4. 15) Prove that (n+1)^2 > n^2 + 2n for all positive integers n. a) A) The statement is unachievable b) B) The statement is true for all n c) C) The statement is not true for any n. d) D) The statement is true for all n except for n = 1.

Mathematical Induction Practice

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