Mathematical induction Test

{"questions":[{"content":"In a proof that 4ⁿ − 1 is divisible by 3, which identity exposes the inductive hypothesis?<br>[[choice-ind-q1]]","widgets":{"choice-ind-q1":{"type":"choice","options":["4ᵏ⁺¹ − 1 = 4(4ᵏ − 1) + 3","4ᵏ⁺¹ − 1 = 4ᵏ + 3","4ᵏ⁺¹ − 1 = 3(4ᵏ − 1)","4ᵏ⁺¹ − 1 = k + 3"],"answer":[0],"explanations":["[M1] Rewrite the next case as 4(4ᵏ − 1) + 3. [A1] Both parts are divisible by 3.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Expand the right-hand side."],"explanation":"[M1] Rewrite the next case as 4(4ᵏ − 1) + 3. [A1] Both parts are divisible by 3.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"For the claim 2 + 4 + ··· + 2n = n(n + 1), the n = 1 base case has value [[input-ind-q2]]","widgets":{"input-ind-q2":{"type":"input","answer":"2"}},"hints":["Evaluate either side at n = 1."],"explanation":"[A1] Both sides equal 2.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["A1"],"follow_through":false}},{"content":"Assume 2 + 4 + ··· + 2k = k(k + 1). After adding 2(k + 1), what factorized form results?<br>[[choice-ind-q3]]","widgets":{"choice-ind-q3":{"type":"choice","options":["(k + 1)(k + 2)","k(k + 2)","2(k + 1)","(k + 1)²"],"answer":[0],"explanations":["[M1] Add 2(k + 1). [A1] k(k + 1) + 2(k + 1) = (k + 1)(k + 2).","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Factor k + 1."],"explanation":"[M1] Add 2(k + 1). [A1] k(k + 1) + 2(k + 1) = (k + 1)(k + 2).","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"A student proves P(1) and P(k) ⇒ P(k + 2). What is the main issue?<br>[[choice-ind-q4]]","widgets":{"choice-ind-q4":{"type":"choice","options":["The argument does not reach every integer from the single base case","The algebra must use decimals","P(1) is never allowed","Induction cannot prove formulas"],"answer":[0],"explanations":["[R1] The step reaches 1, 3, 5, … only; another base case or a k + 1 step is needed.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["List the cases reached from 1."],"explanation":"[R1] The step reaches 1, 3, 5, … only; another base case or a k + 1 step is needed.","assessment":{"paper_mode":"paper_3","calculator_mode":"optional","gdc_skill":"","marks":["R1"],"follow_through":false}},{"content":"Which final line completes a proof beginning at n = 2?<br>[[choice-ind-q5]]","widgets":{"choice-ind-q5":{"type":"choice","options":["Therefore P(n) is true for all integers n ≥ 2 by mathematical induction","P(k + 1) is probably true","The examples are enough","Therefore P(x) holds for all real x"],"answer":[0],"explanations":["[R1] State the method and the exact integer range.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Match the conclusion to the stated domain."],"explanation":"[R1] State the method and the exact integer range.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["R1"],"follow_through":false}}],"mix":1}