// Numbas version: exam_results_page_options
{"name": "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": [{"tags": [], "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>", "variable_groups": [], "ungrouped_variables": ["a1", "d", "small", "large"], "variables": {"large": {"group": "Ungrouped variables", "description": "<p>A large index to compute</p>", "definition": "random(10..50#5)*10", "name": "large", "templateType": "anything"}, "a1": {"group": "Ungrouped variables", "description": "<p>The first&nbsp;term in the sequence</p>", "definition": "random(1..90)", "name": "a1", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "random(3..13 except 10)", "name": "d", "templateType": "anything"}, "small": {"group": "Ungrouped variables", "description": "<p>A&nbsp;small index to compute</p>", "definition": "random(6..10)", "name": "small", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.</p>"}, "parts": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "customName": "", "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>", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "answerSimplification": "basic", "checkingType": "absdiff", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "customMarkingAlgorithm": "", "valuegenerators": [{"value": "", "name": "n"}], "variableReplacements": [], "scripts": {}, "checkVariableNames": false, "failureRate": 1, "marks": 1, "unitTests": [], "customName": "", "answer": "{a1}+(n-1){d}", "showCorrectAnswer": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "showPreview": true}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "customName": "", "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}} = $&nbsp;[[0]]</span></span></p>\n<p><em></em></p>", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 1, "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "customName": "", "notationStyles": ["plain", "en", "si-en"], "showFractionHint": true, "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "minValue": "a1+(small-1)*d", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "maxValue": "a1+(small-1)*d", "useCustomName": false, "showFeedbackIcon": true, "allowFractions": false, "variableReplacements": [], "scripts": {}}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "customName": "", "prompt": "<p>Find the&nbsp;<span color=\"#000000\">$\\var{large}^{\\text{th}}$ term</span></p>\n<p><span color=\"#000000\">$a_{\\var{large}} = $[[0]]</span></p>", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 1, "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "customName": "", "notationStyles": ["plain", "en", "si-en"], "showFractionHint": true, "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "minValue": "a1+(large-1)*d", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "maxValue": "a1+(large-1)*d", "useCustomName": false, "showFeedbackIcon": true, "allowFractions": false, "variableReplacements": [], "scripts": {}}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}}], "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></h4>\n<h4></h4>", "functions": {}, "rulesets": {}, "extensions": [], "name": "Write down and apply the formula for an arithmetic sequence.", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}