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{"name": "Arithmetic progressions: simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "statement": "<p>The sum of the first \\(\\var{n1}\\) terms of an arithmetic progression is \\(\\var{s1}\\) and the \\(\\var{n2}th\\) term of the same series is \\(\\var{T}\\).</p>", "tags": [], "preamble": {"js": "", "css": ""}, "variable_groups": [], "name": "Arithmetic progressions: simultaneous equations", "variables": {"T": {"definition": "random(8..16#1)", "description": "", "name": "T", "templateType": "randrange", "group": "Ungrouped variables"}, "d": {"definition": "({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))", "description": "", "name": "d", "templateType": "anything", "group": "Ungrouped variables"}, "n2": {"definition": "random(5..14#1)", "description": "", "name": "n2", "templateType": "randrange", "group": "Ungrouped variables"}, "n1": {"definition": "random(3..12#1)", "description": "", "name": "n1", "templateType": "randrange", "group": "Ungrouped variables"}, "s1": {"definition": "random(12..60#1)", "description": "", "name": "s1", "templateType": "randrange", "group": "Ungrouped variables"}, "a": {"definition": "{T}-({n2}-1)*{d}", "description": "", "name": "a", "templateType": "anything", "group": "Ungrouped variables"}}, "rulesets": {}, "advice": "<p>Recall the formula for the sum of the first n terms of an arithmetic progression is \\(S_n=\\frac{n}{2}(2a+(n-1)d)\\).</p>\n<p>The sum of the first \\(\\var{n1}\\) terms of an arithmetic progression is \\(\\var{s1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{\\var{n1}}{2}(2a+\\simplify{{n1}-1}d)=\\var{s1}\\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; equation (i)</p>\n<p>The formula for nth term of an arithmetic progression is \\(T_n=a+(n-1)d\\).</p>\n<p>The \\(\\var{n2}th\\) term of the same series is \\(\\var{T}\\)</p>\n<p style=\"padding-left: 30px;\">\\(a+\\simplify{{n2}-1}d=\\var{T}\\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; equation (ii)</p>\n<p>Here we have two simultaneous equations. We can eliminate the \\(a\\) term.</p>\n<p style=\"padding-left: 30px;\">\\(\\var{n1}a+\\simplify{({n1}-1)*{n1}/2}d=\\var{s1}\\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;equation (i)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{n1}a+\\simplify{{n1}*({n2}-1)}d=\\simplify{{n1}*{T}}\\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;equation (ii)*\\(\\var{n1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{({n1}-1)*{n1}/2-{n1}*({n2}-1)}d=\\simplify{{s1}-{n1}*{T}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(d=\\frac{\\simplify{{s1}-{n1}*{T}}}{\\simplify{({n1}-1)*{n1}/2-{n1}*({n2}-1)}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(d=\\simplify{({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\\)</p>\n<p>Using this result and equation (ii) we can find the value for \\(a\\)</p>\n<p style=\"padding-left: 30px;\">\\(a+\\simplify{({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}=\\var{T}\\)</p>\n<p style=\"padding-left: 30px;\">\\(a=\\var{T}-\\simplify{({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\\)</p>\n<p style=\"padding-left: 30px;\">\\(a=\\simplify{{T}-({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solving arithmetic progressions using simultaneous equations</p>"}, "ungrouped_variables": ["n1", "s1", "n2", "T", "d", "a"], "parts": [{"prompt": "<p>Calculate the value of the common difference. &nbsp; \\(d\\) =&nbsp;[[0]]</p>\n<p>Calculate the value of the first term of the series. &nbsp;\\(a\\) =&nbsp;[[1]]</p>", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "{d}", "correctAnswerFraction": false, "precisionPartialCredit": 0, "precisionType": "dp", "showFeedbackIcon": true, "type": "numberentry", "marks": 1, "variableReplacements": [], "mustBeReduced": false, "precision": "1", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{d}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "allowFractions": false, "showPrecisionHint": true, "strictPrecision": false}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "{a}", "correctAnswerFraction": false, "precisionPartialCredit": 0, "precisionType": "dp", "showFeedbackIcon": true, "type": "numberentry", "marks": 1, "variableReplacements": [], "mustBeReduced": false, "precision": "1", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{a}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "allowFractions": false, "showPrecisionHint": true, "strictPrecision": false}], "variableReplacements": [], "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "scripts": {}}], "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/"}]}