// Numbas version: exam_results_page_options
{"name": "Understanding $\\sum$ notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>a) $(1/2)*\\var{upsum[0]}*\\var{coeff[0]}*(1+\\var{upsum[0]})$, here we used that the sum of the first $n$ terms in an arithmetic sequence is $\\frac{n(a+l)}{2}$ where $a$ is the first term and $l$ is the $n$<sup>th</sup> term. We use this formula for&nbsp;b), d) and e) too.</p>\n<p>For the questions involving powers, recall the useful formulae:</p>\n<ol>\n<li>The sum of the first&nbsp;<i>n</i>&nbsp;square numbers is equal to&nbsp;<img src=\"http://pirate.shu.edu/~wachsmut/ira/symbols/sum_sqr.gif\" align=\"middle\"/></li>\n<li>The sum of the first&nbsp;<i>n</i>&nbsp;cubic numbers is equal to&nbsp;<img src=\"http://pirate.shu.edu/~wachsmut/ira/symbols/sum_cub.gif\" align=\"middle\"/></li>\n</ol>\n<p>In some questions it will be helpful to note that $\\sum\\limits_{n=5}^{10}f(n)=\\sum\\limits_{n=1}^{10}f(n)-\\sum\\limits_{n=1}^{4}f(n)$</p>", "parts": [{"correctAnswerFraction": false, "minValue": "(1/2)*{upsum[0]}*{coeff[0]}*(1+{upsum[0]})", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=1}^\\var{upsum[0]} \\var{coeff[0]}n$</p>", "type": "numberentry", "maxValue": "(1/2)*{upsum[0]}*{coeff[0]}*(1+{upsum[0]})", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}, {"correctAnswerFraction": false, "minValue": "(1/2)*{upsum[1]+1}*(2*{coeff[1]}+{upsum[1]})", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=0}^\\var{upsum[1]} (n+\\var{coeff[1]})$</p>", "type": "numberentry", "maxValue": "(1/2)*{upsum[1]+1}*(2*{coeff[1]}+{upsum[1]})", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}, {"correctAnswerFraction": false, "minValue": "(1/4)*({low[0]}^2)*(({low[0]}+1)^2)", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=1}^\\var{low[0]} n^3$</p>", "type": "numberentry", "maxValue": "(1/4)*({low[0]}^2)*(({low[0]}+1)^2)", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}, {"correctAnswerFraction": false, "minValue": "(1/2)*{upsum[2]}*({upsum[2]}+1)", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=1}^\\var{upsum[2]} n$</p>", "type": "numberentry", "maxValue": "(1/2)*{upsum[2]}*({upsum[2]}+1)", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}, {"correctAnswerFraction": false, "minValue": "(1/2)*({upsum[1]}-1)*(2*(4*{coeff[2]}-{coeff[3]})+({upsum[1]}-2)*{coeff[2]})", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=4}^{\\simplify{{upsum[1]}+2}} (\\var{coeff[2]}n-\\var{coeff[3]})$</p>", "type": "numberentry", "maxValue": "(1/2)*({upsum[1]}-1)*(2*(4*{coeff[2]}-{coeff[3]})+({upsum[1]}-2)*{coeff[2]})", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}, {"correctAnswerFraction": false, "minValue": "((1/6)*({upsum[2]})*(({upsum[2]}+1))*((2*{upsum[2]}+1)))-((1/6)*({low[2]})*(({low[2]}-1))*((2*{low[2]}-1)))", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=\\var{low[2]}}^\\var{upsum[2]} n^2$</p>", "type": "numberentry", "maxValue": "((1/6)*({upsum[2]})*(({upsum[2]}+1))*((2*{upsum[2]}+1)))-((1/6)*({low[2]})*(({low[2]}-1))*((2*{low[2]}-1)))", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}, {"correctAnswerFraction": false, "minValue": "((1/6)*({upsum[3]})*(({upsum[3]}+1))*((2*{upsum[3]}+1)))-((1/6)*({low[1]})*(({low[1]}-1))*((2*{low[1]}-1))) + (1/2)*({upsum[3]})*({upsum[3]}+1) - (1/2)*({low[1]})*({low[1]}-1)", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$\\sum\\limits_{n=\\var{low[1]}}^\\var{upsum[3]} n(n+1)$</p>", "type": "numberentry", "maxValue": "((1/6)*({upsum[3]})*(({upsum[3]}+1))*((2*{upsum[3]}+1)))-((1/6)*({low[1]})*(({low[1]}-1))*((2*{low[1]}-1))) + (1/2)*({upsum[3]})*({upsum[3]}+1) - (1/2)*({low[1]})*({low[1]}-1)", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false}], "variable_groups": [], "ungrouped_variables": ["upsum", "coeff", "pow", "low"], "rulesets": {}, "name": "Understanding $\\sum$ notation", "tags": [], "type": "question", "functions": {}, "preamble": {"css": "", "js": ""}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Evaluate each of these</p>", "variables": {"pow": {"templateType": "anything", "definition": "shuffle(2..4)", "description": "", "group": "Ungrouped variables", "name": "pow"}, "upsum": {"templateType": "anything", "definition": "shuffle(5..8)", "description": "", "group": "Ungrouped variables", "name": "upsum"}, "low": {"templateType": "anything", "definition": "shuffle(2..4)", "description": "", "group": "Ungrouped variables", "name": "low"}, "coeff": {"templateType": "anything", "definition": "shuffle(2..9)", "description": "", "group": "Ungrouped variables", "name": "coeff"}}, "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/"}]}