// Numbas version: exam_results_page_options
{"name": "Simon's copy of Solving for a geometric series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>The sum of the first \\(\\var{n}\\) terms of a geometric series is \\(\\var{s_1}\\) and the sum of the first \\(\\simplify{2*{n}}\\) terms is \\(\\var{s_2}\\).</p>", "metadata": {"description": "<p>Solving for a geometric series</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "preamble": {"js": "", "css": ""}, "functions": {}, "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "name": "Simon's copy of Solving for a geometric series", "variable_groups": [], "rulesets": {}, "parts": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showFeedbackIcon": true, "sortAnswers": false, "marks": 0, "unitTests": [], "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "precisionPartialCredit": 0, "unitTests": [], "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": true, "marks": 1, "showFeedbackIcon": true, "precision": "3", "type": "numberentry", "variableReplacements": [], "minValue": "{r}", "precisionType": "dp", "showCorrectAnswer": true, "strictPrecision": false, "customMarkingAlgorithm": "", "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{r}", "scripts": {}, "mustBeReducedPC": 0, "correctAnswerFraction": false}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "precisionPartialCredit": 0, "unitTests": [], "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": true, "marks": 1, "showFeedbackIcon": true, "precision": "3", "type": "numberentry", "variableReplacements": [], "minValue": "{a}", "precisionType": "dp", "showCorrectAnswer": true, "strictPrecision": false, "customMarkingAlgorithm": "", "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{a}", "scripts": {}, "mustBeReducedPC": 0, "correctAnswerFraction": false}], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}, "prompt": "<p>Determine the value of the common ratio. &nbsp; &nbsp;\\(r\\) =&nbsp;[[0]]</p>\n<p>Calculate the value of the first term. &nbsp; &nbsp;\\(a\\) =&nbsp;[[1]]</p>"}], "advice": "<p>\\(S_{\\var{n}}=\\frac{a(1-r^{\\var{n}})}{1-r}=\\var{s_1}\\)</p>\n<p>\\(S_{\\simplify{2*{n}}}=\\frac{a(1-r^{\\simplify{2*{n}}})}{1-r}=\\var{s_2}\\)</p>\n<p>If we divide one by the other we get:</p>\n<p>\\(\\frac{S_{\\simplify{2*{n}}}}{S_{\\var{n}}}=\\frac{\\frac{a(1-r^{\\simplify{2*{n}}})}{1-r}}{\\frac{a(1-r^{\\var{n}})}{1-r}}=\\frac{\\var{s_2}}{\\var{s_1}}\\)</p>\n<p>\\(\\frac{S_{\\simplify{2*{n}}}}{S_{\\var{n}}}=\\frac{a(1-r^{\\simplify{2*{n}}})}{1-r}\\times\\frac{1-r}{a(1-r^{\\var{n}})}=\\frac{\\var{s_2}}{\\var{s_1}}\\)</p>\n<p>\\(\\frac{1-r^{\\simplify{2*{n}}}}{1-r^{\\var{n}}}=\\frac{\\var{s_2}}{\\var{s_1}}\\)</p>\n<p>\\(\\frac{(1-r^\\var{n})(1+r^{\\var{n}})}{1-r^{\\var{n}}}=\\frac{\\var{s_2}}{\\var{s_1}}\\)</p>\n<p>\\(1+r^{\\var{n}}=\\frac{\\var{s_2}}{\\var{s_1}}\\)</p>\n<p>\\(r^{\\var{n}}=\\frac{\\var{s_2}}{\\var{s_1}}-1\\)</p>\n<p>\\(r^{\\var{n}}=\\simplify{{s_2}/{s_1}-1}\\)</p>\n<p>\\(r=\\simplify{(({s_2})/{s_1}-1)^{1/{n}}}\\)</p>\n<p>\\(r=\\simplify{(({s_2}-{s_1})/{s_1})^{1/{n}}}=\\var{r}\\)</p>\n<p>Recall \\(S_{\\var{n}}=\\frac{a(1-r^{\\var{n}})}{1-r}=\\var{s_1}\\)</p>\n<p>\\(a=\\frac{\\var{s_1}\\times(1-{r})}{1-r^{\\var{n}}}\\)</p>\n<p>Inserting the value for \\(r\\) in this equation gives</p>\n<p>\\(a=\\frac{\\var{s_1}\\times(\\simplify{(1-{r})})}{\\simplify{{1-r^{{n}}}}}\\)</p>\n<p>\\(a=\\var{a}\\)</p>\n<p></p>", "tags": [], "ungrouped_variables": ["n", "s_1", "s_2", "r", "a"], "variables": {"s_1": {"name": "s_1", "definition": "random(12..36#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "r": {"name": "r", "definition": "({s_2}/{s_1}-1)^(1/{n})", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"name": "a", "definition": "{s_1}*(1-{r})/(1-{r}^{n})", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "s_2": {"name": "s_2", "definition": "random(36..50#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "n": {"name": "n", "definition": "random(3..7#2)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}