// Numbas version: exam_results_page_options
{"name": "Arithmetic Series 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "d", "n1", "t1", "n2", "t2", "sqdisc", "disc", "n_val"], "name": "Arithmetic Series 3", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>To find the&nbsp;common difference d&nbsp;we note that the&nbsp;$\\var{n1}$<sup>th</sup>&nbsp;term is $\\var{t1}$ and the $\\var{n2}$<sup>th&nbsp;</sup>term is $\\var{t2}$ so we can find the common difference d by noting $\\var{t2}-\\var{t1}=(\\var{n2}-\\var{n1})\\times d$, hence common difference $= \\var{d}$</p>\n<p></p>\n<p>To find the least positive value for&nbsp;<span class=\"MathJax_Preview\"></span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\"><nobr><span class=\"math\" id=\"MathJax-Span-13\" role=\"math\"><span class=\"mrow\" id=\"MathJax-Span-14\"><span class=\"mi\" id=\"MathJax-Span-15\">n</span></span></span></nobr></span>&nbsp;for which the sum of the first&nbsp;<span class=\"MathJax_Preview\"></span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\"><nobr><span class=\"math\" id=\"MathJax-Span-16\" role=\"math\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mi\" id=\"MathJax-Span-18\">n</span></span></span></nobr></span>&nbsp;terms of the series exceeds 1000, we recall that the sum of an arithmetic series can be found with the formula $S_n = \\frac{n(2a_1 + (n-1)\\times d)}{2}$ where $a_1$ denotes the first term and d denotes the common difference. Now we want to find the smallest positive $n$ such that $S_n &gt; 1000$. This is now a matter of solving a quadratic, which is left as an exercise.&nbsp;</p>", "rulesets": {}, "parts": [{"integerPartialCredit": 0, "prompt": "<p>What is the common difference?</p>\n<p></p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{d}", "minValue": "{d}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>Determine the least positive value for $n$ for which the sum of the first $n$ terms of the series exceeds 1000.</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{n_val}", "minValue": "{n_val}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>In an arithmetic series, the $\\var{n1}$<sup>th&nbsp;</sup>term is $\\var{t1}$ and the $\\var{n2}$<sup>th&nbsp;</sup>term is $\\var{t2}$</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "d": {"definition": "random(2..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "n_val": {"definition": "ceil((d-2a+disc)/(2*d))", "templateType": "anything", "group": "Ungrouped variables", "name": "n_val", "description": ""}, "sqdisc": {"definition": "(2a-d)^2+8000*d", "templateType": "anything", "group": "Ungrouped variables", "name": "sqdisc", "description": ""}, "t2": {"definition": "a+(n2-1)*d", "templateType": "anything", "group": "Ungrouped variables", "name": "t2", "description": ""}, "t1": {"definition": "a+(n1-1)*d", "templateType": "anything", "group": "Ungrouped variables", "name": "t1", "description": ""}, "disc": {"definition": "(sqdisc)^0.5", "templateType": "anything", "group": "Ungrouped variables", "name": "disc", "description": ""}, "n1": {"definition": "random(4..19)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "random(4..19 except n1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}