// Numbas version: exam_results_page_options
{"name": "Solving for a geometric series #3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "statement": "<p>The second term in a geometric series is \\(\\var{t2}\\) and the sum to infinity of the series is \\(\\var{s}\\).</p>\n<p>There are two possible series that possess these attributes.</p>", "rulesets": {}, "tags": [], "ungrouped_variables": ["t2", "s", "r_1", "r_2", "a_1", "a_2"], "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>Calculate the value of the larger common ratio. &nbsp; \\(r\\) =&nbsp;[[0]]</p>\n<p>Determine the value of the first term of the series corresponding to this common ratio. &nbsp;\\(a\\) =&nbsp;[[1]]</p>\n<p>Calculate the value of the smaller common ratio. &nbsp; \\(r\\) =&nbsp;[[2]]</p>\n<p>Determine the value of the first term of the series corresponding to this common ratio. &nbsp;\\(a\\) =&nbsp;[[3]]</p>", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": 1, "precisionType": "dp", "precision": "2", "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "{r_1}", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "maxValue": "{r_1}", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "variableReplacements": [], "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": 1, "precisionType": "dp", "precision": "2", "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "{a_1}", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "maxValue": "{a_1}", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "variableReplacements": [], "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": 1, "precisionType": "dp", "precision": "2", "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "{r_2}", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "maxValue": "{r_2}", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "variableReplacements": [], "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": 1, "precisionType": "dp", "precision": "2", "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "{a_2}", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "maxValue": "{a_2}", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "variableReplacements": [], "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}], "functions": {}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solving for a geometric series</p>"}, "type": "question", "variables": {"r_1": {"group": "Ungrouped variables", "description": "", "definition": "({s}+sqrt({s}^2-4*{s}*{t2}))/(2*{s})", "name": "r_1", "templateType": "anything"}, "t2": {"group": "Ungrouped variables", "description": "", "definition": "random(1..9#1)", "name": "t2", "templateType": "randrange"}, "r_2": {"group": "Ungrouped variables", "description": "", "definition": "({s}-sqrt({s}^2-4*{s}*{t2}))/(2*{s})", "name": "r_2", "templateType": "anything"}, "a_2": {"group": "Ungrouped variables", "description": "", "definition": "{t2}/{r_2}", "name": "a_2", "templateType": "anything"}, "a_1": {"group": "Ungrouped variables", "description": "", "definition": "{t2}/{r_1}", "name": "a_1", "templateType": "anything"}, "s": {"group": "Ungrouped variables", "description": "", "definition": "random(36..50#1)", "name": "s", "templateType": "randrange"}}, "advice": "<p>The second term of a geometric series is given by the formula \\(T_2=ar\\) and the sum to infinity of a geometric series is \\(S_\\infty=\\frac{a}{1-r}\\)</p>\n<p style=\"padding-left: 30px;\">\\(T_2=ar=\\var{t2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(a=\\frac{\\var{t2}}{r}\\)</p>\n<p>We can substitute this in for \\(a\\) in the second equation</p>\n<p style=\"padding-left: 30px;\">\\(S_\\infty=\\frac{a}{1-r}=\\var{s}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{\\frac{\\var{t2}}{r}}{1-r}=\\var{s}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{\\var{t2}}{r}=\\var{s}(1-{r})\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{\\var{t2}}{r}=\\var{s}-\\var{s}{r}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{t2}=\\var{s}r-\\var{s}r^2\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{s}r^2-\\var{s}r+\\var{t2}=0\\)&nbsp;</p>\n<p>This is a quadratic equation which we can solve by formula.</p>\n<p style=\"padding-left: 30px;\">\\(r=\\frac{\\var{s}\\pm \\sqrt{(-\\var{s})^2-4*(\\var{s})*(\\var{t2})}}{2*(\\var{s})}\\)</p>\n<p style=\"padding-left: 30px;\">\\(r=\\frac{\\var{s}+\\sqrt{\\simplify{{s}^2-4*{s}*{t2}}}}{\\simplify{2*{s}}}\\) &nbsp; or &nbsp; &nbsp;\\(r=\\frac{\\var{s}-\\sqrt{\\simplify{{s}^2-4*{s}*{t2}}}}{\\simplify{2*{s}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(r=\\frac{\\var{s}+\\simplify{({s}^2-4*{s}*{t2})^0.5}}{\\simplify{2*{s}}}\\) &nbsp; &nbsp;or &nbsp; &nbsp;\\(r=\\frac{\\var{s}-\\simplify{({s}^2-4*{s}*{t2})^0.5}}{\\simplify{2*{s}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(r=\\) {({s}+({s}^2-4*{s}*{t2})^0.5)/(2*{s})} &nbsp; or &nbsp; &nbsp;\\(r=\\) {({s}-({s}^2-4*{s}*{t2})^0.5)/(2*{s})}</p>\n<p>\\(a=\\frac{\\var{t2}}{r}\\)</p>\n<p style=\"padding-left: 30px;\">\\(a=\\) {(2*{s}*{t2})/({s}+({s}^2-4*{s}*{t2})^0.5)} &nbsp; &nbsp;or &nbsp; &nbsp;\\(a=\\) {(2*{s}*{t2})/({s}-({s}^2-4*{s}*{t2})^0.5)}</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p></p>", "extensions": [], "name": "Solving for a geometric series #3", "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}