// Numbas version: exam_results_page_options {"name": "Sorting Maths Induction steps", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Sorting Maths Induction steps", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Suppose we have the proposition that states the $(n^3 - n)$ is always divisible by 3, for all positive integers $n$.

$3|(n^3-n), \\;\\;for\\;all\\;n \\geq 1$

", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Identify correctly the step-by-step procedure to prove this proposition using the First Principle of Mathematical Induction.

Basis Step: when $P(1)$

[[0]]

Thus P(1) is true

Inductive step:

The inductive hypothesis is

[[1]]

Assuming $P(k)$ is true,then for $P(k+1)$

we get [[2]]

= [[3]]

= [[4]]

= [[5]]

= [[6]]

Since $P(k+1)$ is also a multiple of 3, hence we have proven the proposition to be valid using Mathematical Induction.

", "gaps": [{"type": "1_n_2", "useCustomName": true, "customName": "Step 01", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["1", "0", 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Step 02", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["0", "1", 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Step 03", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["0", 0, "1", 0, 0, 0, 0], "distractors": ["", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Step 04", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["0", 0, 0, "1", 0, 0, 0], "distractors": ["", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Step 05", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["0", 0, 0, 0, "1", 0, 0], "distractors": ["", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Step 06", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["0", 0, 0, 0, 0, "1", 0], "distractors": ["", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Step 07", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": false, "choices": ["n³ - n = 1³ - 1 = 0*3", "(k³-k) = 3m, for some integers k and P(k) \u279e P(k+1)", "(n+1)³ - (n-1) = (n^³+3n^²+3n+1) - (n+1)", "n^³-n+3(n^²+n)", "P(k)+3(n^²+n)", "3m+3(n^²+n)", "3(n^²+n+m)"], "matrix": ["0", 0, 0, 0, 0, 0, "1"], "distractors": ["", "", "", "", "", "", ""]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "T LIM", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18062/"}]}]}], "contributors": [{"name": "T LIM", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18062/"}]}