// Numbas version: exam_results_page_options
{"name": "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": [{"metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solving for a geometric series</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}*\\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}*(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}*(\\simplify{(1-{r})})}{\\simplify{{1-r^{{n}}}}}\\)</p>\n<p>\\(a=\\var{a}\\)</p>\n<p></p>", "tags": [], "functions": {}, "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>", "preamble": {"js": "", "css": ""}, "rulesets": {}, "variables": {"n": {"definition": "random(2..6#1)", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "{s_1}*(1-{r})/(1-{r}^{n})", "templateType": "anything", "name": "a", "group": "Ungrouped variables", "description": ""}, "s_2": {"definition": "random(36..50#1)", "templateType": "randrange", "name": "s_2", "group": "Ungrouped variables", "description": ""}, "s_1": {"definition": "random(12..36#1)", "templateType": "randrange", "name": "s_1", "group": "Ungrouped variables", "description": ""}, "r": {"definition": "({s_2}/{s_1}-1)^(1/{n})", "templateType": "anything", "name": "r", "group": "Ungrouped variables", "description": ""}}, "extensions": [], "ungrouped_variables": ["n", "s_1", "s_2", "r", "a"], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Solving for a geometric series", "parts": [{"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "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>", "type": "gapfill", "marks": 0, "gaps": [{"minValue": "{r}", "precision": "2", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{r}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "mustBeReduced": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"minValue": "{a}", "precision": "2", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{a}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "mustBeReduced": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0}], "variableReplacementStrategy": "originalfirst"}], "variable_groups": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}