// Numbas version: exam_results_page_options
{"name": "Find point when a sequence becomes strictly decreasing", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(c)", "name": "n", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t^(1/k)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a/b", "name": "c", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-a", "name": "b", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.5..0.95 # 0.05)", "name": "t", "description": ""}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..20)", "name": "k", "description": ""}}, "ungrouped_variables": ["a", "b", "c", "n", "t", "k"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "name": "Find point when a sequence becomes strictly decreasing", "showQuestionGroupNames": false, "variable_groups": [], "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "integerAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "{n}", "integerPartialCredit": 0, "marks": "8", "maxValue": "{n}"}], "type": "gapfill", "prompt": "<p>Find the smallest integer $N$ such that $x_{m+1} \\leq x_m$ for all $m \\geq N$.&nbsp;</p>\n<p>The smallest integer is [[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let $\\{x_n\\}$ be the sequence given by&nbsp;</p>\n<p>\\[x_n=n^\\var{k} \\var{t}^n\\]</p>", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "limits", "MAS2224", "query", "sequences", "taking the limit", "tested1", "udf"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$x_n=n^k t^n$ where $k$ is a positive integer and $t$ a real number with $0 &lt; t&lt;1$.&nbsp;Find the smallest integer $N$ such that $(m+1)^k t^{m+1} \\leq m^k t^m$ for all $m \\geq N$.&nbsp;</p>"}, "functions": {"chcop": {"type": "number", "language": "jme", "definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "parameters": [["a", "number"], ["b", "number"]]}}, "advice": "<p>The condition $x_{m+1} \\leq x_m$ is $(m+1)^\\var{k} \\var{t}^{m+1} \\leq m^\\var{k} \\var{t}^m$. This can be shown to be equivalent to $m \\geq \\dfrac{\\var{t}^{1/\\var{k}}}{1-\\var{t}^{1/\\var{k}}}=\\var{c} $. We take $N$ to be the smallest integer $\\geq \\var{c} $, so $N=\\var{n}$. Then $m \\geq \\var{c}$ for all $m \\geq \\var{n}$ and therefore $x_{m+1} \\leq x_m$ for all $m \\geq \\var{n}$.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}