// Numbas version: exam_results_page_options
{"name": "Geometric $\\sum$ questions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["upsum", "mult", "num", "denom", "r", "mult2", "r2"], "name": "Geometric $\\sum$ questions 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>We will look at solutions for a). The method for b) is exactly the same.</p>\n<p>To find the first term, plug in i=0. That is&nbsp;$\\frac{1}{\\var{mult2}} \\times (\\var{r2})^{0}=\\frac{1}{\\var{mult2}}\\times 1 = \\frac{1}{\\var{mult2}}$</p>\n<p>To find the common ratio, we look at the $(i+1)$<sup>th&nbsp;</sup>term divided by the $i$<sup>th</sup> term. That is $\\frac{\\frac{1}{\\var{mult2}}\\times&nbsp;\\simplify{{r2}}^{i}}{\\frac{1}{\\var{mult2}}\\times&nbsp;\\simplify{{r2}}^{i-1}} = \\var{r2}$</p>\n<p>To find the value of the sum, we apply the known formula for a geometric series for finding the sum of the first $n$ terms. That is $S_n = \\frac{a_1(1-r^n)}{1-r}$, where $a_1$ is the first term and $r$ is the common ratio. In our case we have&nbsp;$S_n = \\frac{\\frac{1}{\\var{mult2}}(1-\\var{r2}^n)}{1-\\var{r2}}$</p>", "rulesets": {}, "parts": [{"prompt": "<p>For $\\sum\\limits_{i=0}^\\var{upsum} \\frac{1}{\\var{mult2}} \\times (\\var{r2})^{i}$</p>\n<p></p>\n<p>What is the first term?&nbsp;[[0]]</p>\n<p>What is the common ratio?&nbsp;[[1]]</p>\n<p>What is the value of the sum?&nbsp;[[2]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "1/{mult2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "{r2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "1/{mult2}(1-{r2}^({upsum}+1))/(1-{r2})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>For the following geometric series</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"num": {"definition": "random(2..5 except mult)", "templateType": "anything", "group": "Ungrouped variables", "name": "num", "description": ""}, "upsum": {"definition": "random(10..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "upsum", "description": ""}, "r2": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "r": {"definition": "num/denom", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "denom": {"definition": "random(2..5 except mult except num)", "templateType": "anything", "group": "Ungrouped variables", "name": "denom", "description": ""}, "mult2": {"definition": "random(2..5 except r2)", "templateType": "anything", "group": "Ungrouped variables", "name": "mult2", "description": ""}, "mult": {"definition": "random(2..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "mult", "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/"}]}