// Numbas version: finer_feedback_settings
{"name": "Recursively defined sequences.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"s": {"definition": "2^n-1", "type": "number", "parameters": [["n", "number"]], "language": "jme"}, "t": {"definition": "3+4*(n-1)", "type": "number", "parameters": [["n", "number"]], "language": "jme"}}, "ungrouped_variables": [], "name": "Recursively defined sequences.", "tags": [], "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>Given the recursively defined sequence:</p>\n<p>$t(1)=3$,</p>\n<p>$t(n)=t(n-1)+4,\\;\\;(n&gt;1)$.</p>\n<p>Find the following:</p>\n<p>$t(2)=\\;$[[0]], $t(3)=\\;$[[1]], $t(4)=\\;$[[2]].</p>\n<p>Express $t(n)$ as a non-recursive function of $n$:</p>\n<p>$t(n)=\\;$[[3]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "t(2)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "t(3)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "t(4)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"notallowed": {"message": "<p>Your definition should not include the function $t$.</p>", "showStrings": false, "strings": ["t"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "3+4*(n-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Given the sequence $s(n)=2^n-1,\\;\\;\\;n\\in \\mathbb{N}.$</p>\n<p>Find a recursive definition of $s(n)$ in terms of $s(n-1)$:</p>\n<p>$s(1)=\\;$[[0]],</p>\n<p>$s(n)=\\;$[[1]],$\\;\\;\\;n \\gt 1$.</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "1", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "musthave": {"message": "<p>You must define $s(n)$ in terms of $s(n-1)$.</p>", "showStrings": false, "strings": ["s"], "partialCredit": 0}, "scripts": {}, "answer": "2^n-1", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {}, "metadata": {"notes": "", "description": "<p>a) Given a recursively defined sequence $t(n)$, find $t(n)$ as a function of $n$.</p>\n<p>b) Given a sequence $s(n),\\;n \\in \\mathbb{N}$, find a recursive definition of $s(n)$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}