// 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": [{"variables": {"n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(4..19)", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(4..19 except n1)", "description": "", "templateType": "anything"}, "sqdisc": {"name": "sqdisc", "group": "Ungrouped variables", "definition": "(2a-d)^2+8000*d", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "n_val": {"name": "n_val", "group": "Ungrouped variables", "definition": "ceil((d-2a+disc)/(2*d))", "description": "", "templateType": "anything"}, "disc": {"name": "disc", "group": "Ungrouped variables", "definition": "(sqdisc)^0.5", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2..9 except a)", "description": "", "templateType": "anything"}, "t1": {"name": "t1", "group": "Ungrouped variables", "definition": "a+(n1-1)*d", "description": "", "templateType": "anything"}, "t2": {"name": "t2", "group": "Ungrouped variables", "definition": "a+(n2-1)*d", "description": "", "templateType": "anything"}}, "showQuestionGroupNames": false, "ungrouped_variables": ["a", "d", "n1", "t1", "n2", "t2", "sqdisc", "disc", "n_val"], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Arithmetic Series 3", "parts": [{"allowFractions": false, "integerAnswer": true, "integerPartialCredit": 0, "maxValue": "{d}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFraction": false, "showCorrectAnswer": true, "prompt": "<p>What is the common difference?</p>\n<p></p>", "showPrecisionHint": false, "scripts": {}, "minValue": "{d}", "marks": "1", "type": "numberentry"}, {"allowFractions": false, "integerAnswer": true, "integerPartialCredit": 0, "maxValue": "{n_val}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFraction": false, "showCorrectAnswer": true, "prompt": "<p>Determine the least positive value for $n$ for which the sum of the first $n$ terms of the series exceeds 1000.</p>", "showPrecisionHint": false, "scripts": {}, "minValue": "{n_val}", "marks": 1, "type": "numberentry"}], "rulesets": {}, "variable_groups": [], "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>", "metadata": {"description": "", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "tags": [], "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>", "functions": {}, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "type": "question", "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}