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Solving for a geometric series
", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "preamble": {"css": "", "js": ""}, "advice": "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}\\)
\n\\(T_2=ar=\\var{t2}\\)
\n\\(a=\\frac{\\var{t2}}{r}\\)
\nWe can substitute this in for \\(a\\) in the second equation
\n\\(S_\\infty=\\frac{a}{1-r}=\\var{s}\\)
\n\\(\\frac{\\frac{\\var{t2}}{r}}{1-r}=\\var{s}\\)
\n\\(\\frac{\\var{t2}}{r}=\\var{s}(1-{r})\\)
\n\\(\\frac{\\var{t2}}{r}=\\var{s}-\\var{s}{r}\\)
\n\\(\\var{t2}=\\var{s}r-\\var{s}r^2\\)
\n\\(\\var{s}r^2-\\var{s}r+\\var{t2}=0\\)
\nThis is a quadratic equation which we can solve by formula.
\n\\(r=\\frac{\\var{s}\\pm \\sqrt{(-\\var{s})^2-4\\times(\\var{s})\\times(\\var{t2})}}{2\\times(\\var{s})}\\)
\n\\(r=\\frac{\\var{s}+\\sqrt{\\simplify{{s}^2-4*{s}*{t2}}}}{\\simplify{2*{s}}}\\) or \\(r=\\frac{\\var{s}-\\sqrt{\\simplify{{s}^2-4*{s}*{t2}}}}{\\simplify{2*{s}}}\\)
\n\\(r=\\frac{\\var{s}+\\simplify{({s}^2-4*{s}*{t2})^0.5}}{\\simplify{2*{s}}}\\) or \\(r=\\frac{\\var{s}-\\simplify{({s}^2-4*{s}*{t2})^0.5}}{\\simplify{2*{s}}}\\)
\n\\(r=\\) {({s}+({s}^2-4*{s}*{t2})^0.5)/(2*{s})} or \\(r=\\) {({s}-({s}^2-4*{s}*{t2})^0.5)/(2*{s})}
\n\\(a=\\frac{\\var{t2}}{r}\\)
\n\\(a=\\) {(2*{s}*{t2})/({s}+({s}^2-4*{s}*{t2})^0.5)} or \\(a=\\) {(2*{s}*{t2})/({s}-({s}^2-4*{s}*{t2})^0.5)}
\n\n\n\n\n", "tags": [], "functions": {}, "parts": [{"sortAnswers": false, "marks": 0, "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "gaps": [{"precisionType": "dp", "unitTests": [], "correctAnswerFraction": false, "type": "numberentry", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "minValue": "{r_1}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": true, "strictPrecision": false, "maxValue": "{r_1}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionType": "dp", "unitTests": [], "correctAnswerFraction": false, "type": "numberentry", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "minValue": "{a_1}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": true, "strictPrecision": false, "maxValue": "{a_1}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionType": "dp", "unitTests": [], "correctAnswerFraction": false, "type": "numberentry", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "minValue": "{r_2}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": true, "strictPrecision": false, "maxValue": "{r_2}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionType": "dp", "unitTests": [], "correctAnswerFraction": false, "type": "numberentry", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "scripts": {}, "minValue": "{a_2}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": true, "strictPrecision": false, "maxValue": "{a_2}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}], "type": "gapfill", "prompt": "Calculate the value of the larger common ratio. \\(r\\) = [[0]]
\nDetermine the value of the first term of the series corresponding to this common ratio. \\(a\\) = [[1]]
\nCalculate the value of the smaller common ratio. \\(r\\) = [[2]]
\nDetermine the value of the first term of the series corresponding to this common ratio. \\(a\\) = [[3]]
", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "rulesets": {}, "ungrouped_variables": ["t2", "s", "r_1", "r_2", "a_1", "a_2"], "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"s": {"templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(36..50#1)", "name": "s"}, "t2": {"templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..9#1)", "name": "t2"}, "a_2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{t2}/{r_2}", "name": "a_2"}, "r_1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "({s}+sqrt({s}^2-4*{s}*{t2}))/(2*{s})", "name": "r_1"}, "r_2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "({s}-sqrt({s}^2-4*{s}*{t2}))/(2*{s})", "name": "r_2"}, "a_1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{t2}/{r_1}", "name": "a_1"}}, "name": "Simon's copy of Solving for a geometric series #3", "variable_groups": [], "statement": "The second term in a geometric series is \\(\\var{t2}\\) and the sum to infinity of the series is \\(\\var{s}\\).
\nThere are two possible series that possess these attributes.
", "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/"}]}