// Numbas version: exam_results_page_options
{"name": "Calcular la suma parcial de una sucesi\u00f3n aritm\u00e9tica", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"partial_sum": {"group": "Ungrouped variables", "definition": "num_terms*(first_term+last_term)/2", "name": "partial_sum", "templateType": "anything", "description": "<p>Sum of the terms in the sequence.</p>"}, "first_term": {"group": "Ungrouped variables", "definition": "m*first_index", "name": "first_term", "templateType": "anything", "description": "<p>The first term in the sequence.</p>"}, "num_terms": {"group": "Ungrouped variables", "definition": "random(11..30)", "name": "num_terms", "templateType": "anything", "description": "<p>The number of terms in the sequence.</p>"}, "m": {"group": "Ungrouped variables", "definition": "random(2..10)", "name": "m", "templateType": "anything", "description": "<p>Common difference between terms</p>"}, "last_term": {"group": "Ungrouped variables", "definition": "m*last_index", "name": "last_term", "templateType": "anything", "description": "<p>The last term in the sequence.</p>"}, "last_index": {"group": "Ungrouped variables", "definition": "first_index+num_terms-1", "name": "last_index", "templateType": "anything", "description": "<p>The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.</p>\n<p>This is the index of the last term shown.</p>"}, "first_index": {"group": "Ungrouped variables", "definition": "random(6..14)", "name": "first_index", "templateType": "anything", "description": "<p>The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.</p>\n<p>This is the index of the first term shown.</p>"}}, "advice": "<p>Se nos da la secuencia aritm&eacute;tica</p>\n<p>\\[\\var{m*first_index}, \\var{m*(first_index+1)}, \\var{m*(first_index+1)}, \\ldots, \\var{last_term} \\text{.} \\]</p>\n<h4>a)</h4>\n<p>Sea $n$ el n&uacute;mero de t&eacute;rminos en la secuencia. Entonces $a_n = \\var{last_term}$.</p>\n<p>Para encontrar $n$, debemos reorganizar la f&oacute;rmula para el t&eacute;rmino $n^\\text{th}$&nbsp;</p>\n<p>\\[a_n=a_1+(n-1)d\\text{.}\\]</p>\n<p>El primer t&eacute;rmino es $a_1 = \\var{first_term}$ y la diferencia com&uacute;n es $d = \\var{first_term+m} - \\var{first_term} = \\var{m}$.</p>\n<p>Entonces tenemos</p>\n<p>\\begin{align}<br/>d&amp;=\\var{m} \\text{,} \\\\<br/>a_1&amp;= \\var{first_term} \\text{,} \\\\<br/>a_n&amp;=\\var{last_term} \\text{.}<br/>\\end{align}</p>\n<p>Ahora sustituimos estos valores en la f&oacute;rmula del t&eacute;rmino&nbsp; $n^\\text{th}$ y reorganizamos para encontrar $n$.</p>\n<p>\\begin{align}<br/>\\var{last_term}&amp;=\\var{first_term}+\\var{m}(n-1) \\\\<br/>\\var{last_term}-\\var{first_term}&amp;=\\var{m}(n-1) \\\\<br/>\\frac{\\var{last_term-first_term}}{\\var{m}}&amp;=(n-1) \\\\<br/>n&amp;=\\var{(last_term-first_term)/m}+1 = \\var{num_terms}\\text{.}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>La suma parcial de los primeros $n$ t&eacute;rminos de una secuencia aritm&eacute;tica est&aacute; dada por</p>\n<p>\\[\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\\]</p>\n<p>Sabemos de la parte&nbsp;<strong>a) </strong>que</p>\n<p>\\begin{align}<br/>n&amp;= \\var{num_terms} \\text{,} \\\\<br/>a_1&amp;= \\var{first_term} \\text{,} \\\\<br/>a_n&amp;= \\var{last_term} \\text{.}<br/>\\end{align}</p>\n<p>Sustituimos estos valores en la f&oacute;rmula, obteniendo</p>\n<p>\\begin{align}<br/>\\sum\\limits_{i=1}^n{a_i}&amp;=\\frac{n}{2}(a_1+a_n)\\\\<br/>&amp;= \\frac{\\var{num_terms}}{2}(\\var{first_term}+\\var{last_term})\\\\<br/>&amp;= \\simplify[]{ ({num_terms}*({first_term+last_term}))/2} \\\\<br/>&amp;= \\var{partial_sum} \\text{.}<br/>\\end{align}</p>", "metadata": {"description": "<p>Dados los primeros y &uacute;ltimos t&eacute;rminos de una secuencia aritm&eacute;tica finita, calcule el n&uacute;mero de elementos y luego la suma de la secuencia.</p>\n<p>Cada parte se divide en pasos, con la f&oacute;rmula dada.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "parts": [{"sortAnswers": false, "steps": [{"unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>La f&oacute;rmula para el $n^\\text{th}$ t&eacute;rmino en una secuencia aritm&eacute;tica es</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\[a_n=a_1+(n-1)d\\text{,}\\]</p>\n<p>donde</p>\n<ul>\n<li>$d$ es la diferencia com&uacute;n entre t&eacute;rminos consecutivos;</li>\n<li>$a_1$ es el primer t&eacute;rmino de la secuencia;</li>\n<li>$a_n$ es el $n^\\text{th}$ t&eacute;rmino en la&nbsp; secuencia.</li>\n</ul>", "marks": 0, "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"mustBeReducedPC": 0, "minValue": "first_term", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "type": "numberentry", "allowFractions": false, "unitTests": [], "scripts": {}, "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>Para esta secuencia aritm&eacute;tica, &iquest;cu&aacute;l es el valor de $a_1$?</p>", "marks": "0.2", "maxValue": "first_term", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"mustBeReducedPC": 0, "minValue": "m", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "type": "numberentry", "allowFractions": false, "unitTests": [], "scripts": {}, "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>Cual es el valor de $d$?</p>", "marks": "0.2", "maxValue": "m", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "unitTests": [], "scripts": {}, "gaps": [{"mustBeReducedPC": 0, "maxValue": "num_terms", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "mustBeReduced": false, "unitTests": [], "scripts": {}, "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "minValue": "num_terms", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>&iquest;Cu&aacute;ntos t&eacute;rminos hay en esta secuencia?</p>\n<p>$n =$ [[0]]</p>", "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "stepsPenalty": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "steps": [{"unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>La suma de una secuencia aritm&eacute;tica $a_1, a_2, \\ldots, a_n$ es calculada usando la f&oacute;rmula</p>\n<p>\\[ \\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n) \\]</p>\n<p>donde</p>\n<ul>\n<li>$a_1$ es primer t&eacute;rmino;</li>\n<li>$a_n$ es el $n^{th}$&nbsp;t&eacute;rmino;</li>\n<li>$n$ es el n&uacute;mero de t&eacute;rminos.</li>\n</ul>", "marks": 0, "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": ""}], "type": "gapfill", "unitTests": [], "scripts": {}, "gaps": [{"mustBeReducedPC": 0, "maxValue": "partial_sum", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "mustBeReduced": false, "unitTests": [], "scripts": {}, "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "minValue": "partial_sum", "marks": "2", "allowFractions": false, "correctAnswerFraction": false, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>Encuentra la suma de la secuencia incluyendo el t&eacute;rmino $\\var{last_term}$.</p>\n<p>[[0]]</p>", "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "stepsPenalty": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "extensions": [], "rulesets": {}, "functions": {}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "name": "Calcular la suma parcial de una sucesi\u00f3n aritm\u00e9tica", "statement": "<p>Dada la siguiente secuencia aritm&eacute;tica:</p>\n<p style=\"text-align: center;\">$\\var{m*first_index}, \\var{m*(first_index+1)},&nbsp;\\var{m*(first_index+2)}, \\ldots, \\var{last_term}.$</p>", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["m", "first_index", "last_index", "num_terms", "first_term", "last_term", "partial_sum"], "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}