// 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": [{"metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "
Solving for a geometric series
"}, "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*(\\var{s})*(\\var{t2})}}{2*(\\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", "variable_groups": [], "parts": [{"scripts": {}, "showFeedbackIcon": true, "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]]
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\nThere are two possible series that possess these attributes.
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