// Numbas version: finer_feedback_settings
{"name": "Vollst\u00e4ndige Induktion", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vollst\u00e4ndige Induktion", "tags": [], "metadata": {"description": "<p>Ein einfacher, gef&uuml;hrter Induktionsbeweis</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "<p><strong>Zu zeigen:</strong> F&uuml;r alle $n\\in\\mathbb{N}$ gilt: $\\,\\var{a}^{n+1} + \\var{a + 1}^{2n&minus;1}$ ist durch $\\var{a^2 + a + 1}$ teilbar.</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 870px; width: 0px; top: 31px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 855px; width: 50px; top: 16px; height: 20px; font-size: 10px; color: black;\"></span></p>", "advice": "<p>Induktionsanfang: Wir setzen 1 f&uuml;r $n$ ein:&nbsp;$A(1):\\ \\,\\var{a}^{1+1} + \\var{a + 1}^{2\\cdot 1&minus;1}=\\var{a}^{2} + \\var{a + 1}=\\var{a1}$, was offensichtlich durch $\\var{a1}$ teilbar ist.</p>\n<hr/>\n<p>Induktionsbehauptung: Wir setzen zun&auml;chst $(n+1)$ f&uuml;r $n$ ein:&nbsp;$A(n+1):\\ \\,\\var{a}^{(n+1)+1} + \\var{a + 1}^{2\\cdot(n+1)&minus;1}=\\var{a}^{n+2} + \\var{a + 1}^{2n+1}$</p>\n<p>Die Umformungsschritte nutzen dann die Potenzrechenregel&nbsp;$a^{(b+c)}=a^b+a^c$ und&nbsp;das Distibutivgesetz $(a+b)\\cdot c=a\\cdot c+b\\cdot c$&nbsp;</p>\n<p>Womit sich ergibt:</p>\n<p style=\"text-align: center;\">$A(n+1):\\ \\,\\var{a}^{n+2} +\\var{a + 1}^{2n+1}=\\var{k1}\\cdot\\,\\var{a}^{n+1} +\\var{k2}\\cdot\\,\\var{a + 1}^{2n&minus;1}$</p>\n<p style=\"text-align: center;\"><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 559px; width: 0px; top: 99px; height: 0px;\"></span>$=\\var{k1}\\cdot\\,\\var{a}^{n+1} +\\var{k1}\\cdot\\,\\var{a + 1}^{2n&minus;1}+\\var{k2-k1}\\cdot\\,\\var{a + 1}^{2n&minus;1}=\\var{k3}\\cdot\\left(\\var{a}^{n+1} + \\var{a + 1}^{2n&minus;1}\\right)+\\var{k4}\\cdot\\,\\var{a + 1}^{2n&minus;1}$</p>\n<p>Der erste Summand ist gem&auml;&szlig; Induktionsvoraussetzung durch $\\var{a1}$ teilbar, der zweite Summand, weil er ein Vielfaches von $\\var{a1}$ ist. Die Summe ist dann auch durch $\\var{a1}$ teilbar.</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "a^2+a+1", "description": "", "templateType": "anything"}, "k2": {"name": "k2", "group": "Ungrouped variables", "definition": "(a+1)^2", "description": "", "templateType": "anything"}, "k1": {"name": "k1", "group": "Ungrouped variables", "definition": "a", "description": "", "templateType": "anything"}, "k3": {"name": "k3", "group": "Ungrouped variables", "definition": "a", "description": "", "templateType": "anything"}, "k4": {"name": "k4", "group": "Ungrouped variables", "definition": "a^2+a+1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "a1", "k2", "k1", "k3", "k4"], "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": "<p><strong>Induktionsanfang $A(1)$:</strong></p>\n<p>Durch Einsetzen ergibt sich:&nbsp;[[0]], was ein ganzzahliges Vielfaches von $\\var{a^2 + a + 1}$ ist.</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "IA", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1", "maxValue": "a1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"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": "<p><strong>Induktionsschluss $A(n)\\Rightarrow A(n+1)$:</strong></p>\n<p>Es ist zu zeigen, dass unter der <strong>Induktionsvoraussetzung</strong></p>\n<p style=\"text-align: center;\">$A(n):\\ \\,\\var{a}^{n+1} + \\var{a + 1}^{2n&minus;1}$ ist durch $\\var{a^2 + a + 1}$ teilbar (f&uuml;r beliebiges aber festes $n$)</p>\n<p>die <strong>Induktionsbehauptung</strong></p>\n<p style=\"text-align: center;\">$A(n+1):\\,$ [[0]] ist durch $\\var{a^2 + a + 1}$ teilbar&nbsp;</p>\n<p>richtig ist.</p>\n<p>Wir&nbsp;formen jetzt den Term in der Aussage $A(n+1)$ so um, dass wir den Term aus der Aussage $A(n)$ wieder erkennen, dazu heben wir wie folgt heraus:</p>\n<p style=\"text-align: center;\">$A(n+1):\\,$&nbsp;[[0]]$=$[[1]]$\\cdot\\,\\var{a}^{n+1} +$[[2]]$\\cdot\\,\\var{a + 1}^{2n&minus;1}$</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 559px; width: 0px; top: 99px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 544px; width: 50px; top: 84px; height: 20px; font-size: 10px; color: black;\"></span>als n&auml;chstes m&uuml;ssen wir den gesamten Term $\\var{a}^{n+1} + \\var{a + 1}^{2n&minus;1}$ herausheben:</p>\n<p style=\"text-align: center;\">$\\qquad\\qquad=$[[3]]$\\cdot\\left(\\var{a}^{n+1} + \\var{a + 1}^{2n&minus;1}\\right)+$[[4]]$\\cdot\\,\\var{a + 1}^{2n&minus;1}$</p>\n<p style=\"text-align: left;\">Dieser Term ist durch&nbsp;$\\var{a^2 + a + 1}$ teilbar, weil es der erste&nbsp;Summand wegen der Induktionsvoraussetzung ist, warum es auch der zweite ist, erkennen Sie, wenn Sie korrekt herausgehoben haben. Sind zwei Summanden durch eine Zahl teilbar, dann ist es immer auch deren Summe.</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "IB", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}^(n+2) + {a + 1}^(2n+1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n", "value": ""}]}, {"type": "numberentry", "useCustomName": true, "customName": "k1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "k1", "maxValue": "k1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "eu"}, {"type": "numberentry", "useCustomName": true, "customName": "k2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "k2", "maxValue": "k2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "eu"}, {"type": "numberentry", "useCustomName": true, "customName": "k3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "k3", "maxValue": "k3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "eu"}, {"type": "numberentry", "useCustomName": true, "customName": "k4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "k4", "maxValue": "k4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "eu"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "resources": []}]}], "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}