// Numbas version: exam_results_page_options
{"name": "Ugur's copy of Write down and apply the formula for an arithmetic sequence.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Write down and apply the formula for an arithmetic sequence.", "tags": [], "metadata": {"description": "<p>Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><span color=\"#000000\" style=\"color: #000000;\">In this question, consider the sequence</span></p>\n<p><span color=\"#000000\" style=\"color: #000000;\">\\[ a = \\var{a1}, \\; \\var{a1+d}, \\; \\var{a1+d*2}, \\; \\var{a1+d*3}, \\; \\ldots \\]</span></p>\n<p><span color=\"#000000\" style=\"color: #000000;\"></span>A helpful person has drawn out a table of the terms so far.</p>\n<table>\n<tbody>\n<tr>\n<th>$\\boldsymbol{n}$</th>\n<td>$1$</td>\n<td>$2$</td>\n<td>$3$</td>\n<td>$4$</td>\n<td>$\\ldots$</td>\n</tr>\n<tr>\n<th>$\\boldsymbol{a_n}$</th>\n<td>$\\var{a1}$</td>\n<td>$\\var{a1+d}$</td>\n<td>$\\var{a1+2d}$</td>\n<td>$\\var{a1+3d}$</td>\n<td>$\\ldots$</td>\n</tr>\n</tbody>\n</table>", "advice": "<p>The formula for the $n^\\text{th}$ term, $a_n$, of an arithmetic sequence is</p>\n<p><span color=\"#000000\">\\[ a_n=a_1+(n-1)d \\text{.} \\]</span></p>\n<p><span color=\"#000000\">$a_1$ is the first term, and $d$ is the common difference between adjacent terms.</span></p>\n<h4><span color=\"#000000\">a)</span></h4>\n<p><span color=\"#000000\">In the given sequence, the common difference is $\\var{a1+d} - \\var{a1} = \\var{d}$, and the first term is $\\var{a1}$.</span></p>\n<p><span color=\"#000000\">So, the formula for this sequence is</span></p>\n<p><span color=\"#000000\">\\[ a_n = \\var{a1} + (n-1) \\times \\var{d} \\text{.} \\]</span></p>\n<h4><span color=\"#000000\">b)</span></h4>\n<p>\\[ a_\\var{small} = \\var{a1} + (\\var{small}-1) \\times \\var{d} = \\var{a1+(small-1)*d} \\text{.} \\]</p>\n<h4><span color=\"#000000\">c)</span></h4>\n<p>\\[ a_\\var{large} = \\var{a1} + (\\var{large}-1) \\times \\var{d} = \\var{a1+(large-1)*d} \\text{.} \\]</p>\n<h4><em></em>d)&nbsp;</h4>\n<p>\\[S_n = \\frac{n(a_1 + a_n)}{2}\\]&nbsp;</p>\n<p>Plug in the values to get $S_{\\var{medium}} = \\frac{\\var{medium}(\\var{a1} + \\var{amed})}{2} = \\var{ansd} $</p>\n<h4></h4>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"large": {"name": "large", "group": "Ungrouped variables", "definition": "random(10..50#5)*10", "description": "<p>A large index to compute</p>", "templateType": "anything", "can_override": false}, "small": {"name": "small", "group": "Ungrouped variables", "definition": "random(6..10)", "description": "<p>A&nbsp;small index to compute</p>", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..90)", "description": "<p>The first&nbsp;term in the sequence</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(3..13 except 10)", "description": "", "templateType": "anything", "can_override": false}, "medium": {"name": "medium", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "templateType": "anything", "can_override": false}, "amed": {"name": "amed", "group": "Ungrouped variables", "definition": "a1+(medium-1)*d", "description": "", "templateType": "anything", "can_override": false}, "ansd": {"name": "ansd", "group": "Ungrouped variables", "definition": "(medium*(a1 + amed))/2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "d", "small", "large", "medium", "amed", "ansd"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write out an expression for&nbsp;$a_n$, the $n^{\\text{th}}$ term of the sequence, in terms of $n$.</p>\n<p>$a_n =$&nbsp;[[0]]</p>\n<p><em><span style=\"color: #000000;\" color=\"#000000\"><span style=\"color: #000000;\" color=\"#000000\"></span></span></em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": 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"suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p></p>\n<p><span style=\"color: #000000;\" color=\"#000000\"><span style=\"color: #000000;\" color=\"#000000\">Find the $\\var{small}^{\\text{th}}$ term</span></span></p>\n<p><span style=\"color: #000000;\" color=\"#000000\"><span style=\"color: #000000;\" color=\"#000000\">$a_{\\var{small}} = $&nbsp;[[0]]</span></span></p>\n<p><em></em></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1+(small-1)*d", "maxValue": "a1+(small-1)*d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, 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"variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1+(large-1)*d", "maxValue": "a1+(large-1)*d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the sum of first $\\var{medium}$ terms.</p>\n<p>$S_{\\var{medium}} =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap sum", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "({medium}*({a1} + {amed}))/2", "maxValue": "({medium}*({a1} + {amed}))/2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Ugur Efem", "profile_url": 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