// Numbas version: exam_results_page_options
{"name": "Progresiones aritm\u00e9ticas: Ecuaciones simultaneas", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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>", "ungrouped_variables": ["n1", "s1", "n2", "T", "d", "a"], "rulesets": {}, "tags": [], "statement": "<p><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">La suma de los primeros t&eacute;rminos \\ (\\ var {n1} \\) de una progresi&oacute;n aritm&eacute;tica es \\ (\\ var {s1} \\) y el t&eacute;rmino \\ (\\ var {n2} th \\) de la misma serie es \\ (\\ var {T} \\).</span></span></span></span></p>", "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solving arithmetic progressions using simultaneous equations</p>"}, "parts": [{"unitTests": [], "type": "gapfill", "gaps": [{"showPrecisionHint": true, "strictPrecision": false, "precision": "1", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{d}", "mustBeReduced": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "type": "numberentry", "allowFractions": false, "customMarkingAlgorithm": "", "precisionType": "dp", "variableReplacements": [], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "minValue": "{d}", "showFeedbackIcon": true, "mustBeReducedPC": 0}, {"showPrecisionHint": true, "strictPrecision": false, "precision": "1", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{a}", "mustBeReduced": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "type": "numberentry", "allowFractions": false, "customMarkingAlgorithm": "", "precisionType": "dp", "variableReplacements": [], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "minValue": "{a}", "showFeedbackIcon": true, "mustBeReducedPC": 0}], "sortAnswers": false, "customMarkingAlgorithm": "", "scripts": {}, "marks": 0, "variableReplacements": [], "prompt": "<p><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">Calcula el valor de la diferencia com&uacute;n. </span><span style=\"vertical-align: inherit;\">\\ (d \\) =&nbsp;</span></span><gapfill class=\"mceNonEditable\"><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">Gap </span></span><span class=\"gap-number\"><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">0</span></span></span></gapfill></p>\n<p><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">Calcula el valor del primer t&eacute;rmino de la serie. </span><span style=\"vertical-align: inherit;\">\\ (a \\) =&nbsp;</span></span><gapfill class=\"mceNonEditable\"><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">Gap </span></span><span class=\"gap-number\"><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">1</span></span></span></gapfill></p>", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "variable_groups": [], "name": "Progresiones aritm\u00e9ticas: Ecuaciones simultaneas", "variables": {"n2": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "name": "n2", "definition": "random(5..14#1)"}, "s1": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "name": "s1", "definition": "random(12..60#1)"}, "T": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "name": "T", "definition": "random(8..16#1)"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a", "definition": "{T}-({n2}-1)*{d}"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "d", "definition": "({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))"}, "n1": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "name": "n1", "definition": "random(3..12#1)"}}, "type": "question", "contributors": [{"name": "karl walton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2516/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "karl walton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2516/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}