// Numbas version: finer_feedback_settings
{"name": "Encontrar la f\u00f3rmula para el t\u00e9rmino $n^{\\text {th}}$ de una secuencia lineal", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"type": "gapfill", "showFeedbackIcon": true, "scripts": {}, "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "steps": [{"showCorrectAnswer": true, "prompt": "<p>La f&oacute;rmula para el t&eacute;rmino $n^{\\text{th}}$ de una secuencia aritm&eacute;tica es:&nbsp;\\[a_n = a_1 + (n-1)d, \\]</p>\n<p>donde</p>\n<ul>\n<li>$ a_n$ es el t&eacute;rmino $n ^\\text {th} $;</li>\n<li>$ a_1$ es el primer t&eacute;rmino en la secuencia;</li>\n<li>$ d $ es la diferencia com&uacute;n entre t&eacute;rminos consecutivos.</li>\n</ul>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "type": "information", "scripts": {}, "showFeedbackIcon": true, "unitTests": [], "marks": 0}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "showFeedbackIcon": true, "minValue": "{m[1]*2}", "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "maxValue": "{m[1]*2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "prompt": "<p>Para esta secuencia aritm&eacute;tica, &iquest;qu&eacute; es $ a_1 $?</p>", "unitTests": [], "marks": 1, "mustBeReducedPC": 0}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "showFeedbackIcon": true, "minValue": "{m[1]}", "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "maxValue": "{m[1]}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "prompt": "<p>&iquest;Qu&eacute; es $d$?</p>", "unitTests": [], "marks": 1, "mustBeReducedPC": 0}], "extendBaseMarkingAlgorithm": true, "gaps": [{"expectedVariableNames": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "checkVariableNames": false, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "failureRate": 1, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "showPreview": true, "checkingType": "absdiff", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "{m[1]}*n+{m[1]}", "unitTests": [], "marks": "3"}], "prompt": "<p>$\\var{m[1]*2}, \\var{m[1]*3},&nbsp;\\var{m[1]*4}, \\ldots$</p>\n<p>T&eacute;rmino de lugar&nbsp; $n^\\text{th}$ =&nbsp;[[0]]</p>", "sortAnswers": false, "unitTests": [], "marks": 0}, {"type": "gapfill", "showFeedbackIcon": true, "scripts": {}, "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "steps": [{"showCorrectAnswer": true, "prompt": "<p>La f&oacute;rmula para el t&eacute;rmino $ n^{\\text {th}}$ de una secuencia aritm&eacute;tica es $a_n = a_1 + (n-1) d $<br/>d&oacute;nde<br/>$ a_n $ es el t&eacute;rmino $ n ^ \\ text {th} $;<br/>$ a_1 $ es el primer t&eacute;rmino en la secuencia;<br/>$ d $ es la diferencia com&uacute;n entre t&eacute;rminos consecutivos.</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "type": "information", "scripts": {}, "showFeedbackIcon": true, "unitTests": [], "marks": 0}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "showFeedbackIcon": true, "minValue": "{m[2]*8+2}", "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "maxValue": "{m[2]*8+2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "prompt": "<p>Para esta secuencia aritm&eacute;tica, &iquest;qu&eacute; es $ a_1 $?</p>", "unitTests": [], "marks": 1, "mustBeReducedPC": 0}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "showFeedbackIcon": true, "minValue": "{-m[2]}", "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "maxValue": "{-m[2]}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "prompt": "<p>&iquest;Qu&eacute; es&nbsp;$d$?</p>", "unitTests": [], "marks": 1, "mustBeReducedPC": 0}], "extendBaseMarkingAlgorithm": true, "gaps": [{"expectedVariableNames": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "checkVariableNames": false, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "failureRate": 1, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "showPreview": true, "checkingType": "absdiff", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "-{m[2]}*n+{m[2]*9+2}", "unitTests": [], "marks": "3"}], "prompt": "<p>$\\var{m[2]*8+2}, \\var{m[2]*7+2},&nbsp;\\var{m[2]*6+2}...$</p>\n<p>T&eacute;rmino $n^\\text{th}$ =&nbsp;[[0]]</p>", "sortAnswers": false, "unitTests": [], "marks": 0}], "rulesets": {}, "metadata": {"description": "<p>Dados los primeros tres t&eacute;rminos de una secuencia, da una f&oacute;rmula para el t&eacute;rmino $ n^\\text {th} $.</p>\n<p>En la primera secuencia, $d$ es positivo. En la segunda secuencia, $d$ es negativo.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "name": "Encontrar la f\u00f3rmula para el t\u00e9rmino $n^{\\text {th}}$ de una secuencia lineal", "tags": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "statement": "<p>Una secuencia lineal es una serie de n&uacute;meros que aumenta o disminuye en una cantidad constante en cada paso.</p>\n<p>Encuentre f&oacute;rmulas para el t&eacute;rmino $n^ {\\text{th}} $ para cada una de las siguientes secuencias lineales, donde se dan los valores para $n = 1 \\text {,} 2 \\text {,}3 $:</p>", "variables": {"n": {"group": "Ungrouped variables", "description": "", "name": "n", "templateType": "anything", "definition": "repeat(random(1..4),7)"}, "ci": {"group": "Ungrouped variables", "description": "", "name": "ci", "templateType": "anything", "definition": "repeat(random(6..20),10)"}, "m": {"group": "Ungrouped variables", "description": "", "name": "m", "templateType": "anything", "definition": "repeat(random(2..10),5)"}, "c": {"group": "Ungrouped variables", "description": "", "name": "c", "templateType": "anything", "definition": "repeat(random(3..13 except[10]),8)"}, "b": {"group": "Ungrouped variables", "description": "", "name": "b", "templateType": "anything", "definition": "repeat(random(2..4), 5)"}, "ni": {"group": "Ungrouped variables", "description": "", "name": "ni", "templateType": "anything", "definition": "repeat(random(19..40),10)"}}, "advice": "<p>Ambas secuencias son lineales o aritm&eacute;ticas. Para encontrar f&oacute;rmulas para estas secuencias necesitamos identificar sus primeros t&eacute;rminos y diferencias comunes..</p>\n<h4>a)</h4>\n<p>La f&oacute;rmula para el t&eacute;rmino $n^\\text {th} $ de una secuencia aritm&eacute;tica es:&nbsp;$a_n = a_1 + (n-1) d \\text {.}$</p>\n<p>$ a_1 $ es el primer t&eacute;rmino y $d$ la diferencia com&uacute;n entre t&eacute;rminos consecutivos, que debemos identificar.</p>\n<p>Podemos encontrarlos dibujando una tabla de $a_n$ contra $ n $, y las diferencias entre t&eacute;rminos consecutivos.</p>\n<table style=\"width: 540px; height: 150px; margin-left: auto; margin-right: auto;\" cellspacing=\"250\" cellpadding=\"250\" border=\"1\">\n<tbody>\n<tr>\n<td>$n$</td>\n<td>1</td>\n<td>2</td>\n<td>3</td>\n</tr>\n<tr style=\"height: 29px;\">\n<td style=\"height: 29px;\">$a_n$</td>\n<td style=\"height: 29px;\">$\\pmb{\\var{m[1]*2}}$</td>\n<td style=\"height: 29px;\">$\\var{m[1]*3}$</td>\n<td style=\"height: 29px;\">$\\var{m[1]*4}$</td>\n</tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px;\">Primeras diferencias</td>\n<td style=\"height: 46px;\"></td>\n<td style=\"height: 46px;\">$\\pmb{\\var{m[1]}}$</td>\n<td style=\"height: 46px;\">$\\pmb{\\var{m[1]}}$</td>\n</tr>\n</tbody>\n</table>\n<p>El primer t&eacute;rmino y la diferencia com&uacute;n se han resaltado en negrita; Podemos usar estos para escribir la f&oacute;rmula de la secuencia.</p>\n<p>\\begin{align}<br/>a_n &amp;= a_1+(n-1)d \\\\ <br/>&amp;= \\var{m[1]*2}+(n-1)\\times\\var{m[1]} \\\\<br/>&amp;= \\var{m[1]}n +&nbsp;\\var{m[1]}\\text{.}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>De manera similar a la parte a), podemos identificar $ a_1 $ y $ d $ para esta secuencia dibujando una tabla de $ a_n $ contra $ n $.</p>\n<table style=\"width: 540px; height: 150px; margin-left: auto; margin-right: auto;\" cellspacing=\"250\" cellpadding=\"250\" border=\"1\">\n<tbody>\n<tr>\n<td>$a_n$</td>\n<td>$\\pmb{\\var{m[2]*8+2}}$</td>\n<td>$\\simplify{{m[2]}*7+2}$</td>\n<td>$\\simplify{{m[2]}*6+2}$</td>\n</tr>\n<tr>\n<td>Primeras diferencias</td>\n<td></td>\n<td>$\\pmb{\\var{-m[2]}}$</td>\n<td>$\\pmb{\\var{-m[2]}}$</td>\n</tr>\n</tbody>\n</table>\n<p>El primer t&eacute;rmino y la diferencia com&uacute;n se han resaltado en negrita; Podemos usar estos para formar la f&oacute;rmula para la secuencia.</p>\n<p>\\begin{align}<br/>a_n &amp;=a_1+(n-1)d \\\\ <br/>&amp;=\\var{m[2]*8+2}+(n-1)\\times\\var{-m[2]} \\\\<br/>&amp;=-\\var{m[2]}n +&nbsp;\\var{m[2]*9+2}\\text{.}<br/>\\end{align}</p>", "preamble": {"css": "", "js": ""}, "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/"}], "resources": []}]}], "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/"}]}