// Numbas version: exam_results_page_options
{"name": "Find common difference in arithmetic sequences with gaps", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["Arithmetic Sequences", "Arithmetic sequences", "arithmetic sequences", "common difference", "sequences", "taxonomy"], "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>$\\var{n[0]*index[0]}$, $\\var{n[0]*(index[0]+1)}$, $\\var{n[0]*(index[0]+2)}$, $\\var{n[0]*(index[0]+3)}$</p>\n<p>Common Difference =&nbsp;[[0]]</p>", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "{n[0]}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "{n[0]}", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>$\\var{n[1]*(index[1])}$, $?$,&nbsp;$\\var{n[1]*(index[1]+2)}$, $?$, $\\var{n[1]*(index[1]+4)}$</p>\n<p>Common Difference =&nbsp;[[0]]</p>", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "{n[1]}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "{n[1]}", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>$\\var{n[2]*(index[2])}$, $?$, $?$, $\\var{n[2]*(index[2]+3)}$, $?$, $?$, $\\var{n[2]*(index[2]+6)}$</p>\n<p>Common Difference =&nbsp;[[0]]</p>", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "{n[2]}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "{n[2]}", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}], "variable_groups": [], "ungrouped_variables": ["n", "index"], "variables": {"index": {"group": "Ungrouped variables", "description": "", "definition": "shuffle(5..15 except 10)[0..3]", "name": "index", "templateType": "anything"}, "n": {"group": "Ungrouped variables", "description": "", "definition": "shuffle(3..9)[0..3]", "name": "n", "templateType": "anything"}}, "functions": {}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given sequences with missing terms, find the common difference between terms.</p>"}, "type": "question", "statement": "<p>Find the common differences of the following arithmetic sequences. Some of the terms are missing.</p>", "advice": "<p>In an arithmetic sequence, the difference between two adjecent&nbsp;terms is always the same.</p>\n<p>Call the common difference $d$, and the first term in the sequence $a_0$. Then the sequence goes as follows:</p>\n<p>\\[ a_0, \\; a_0+d, \\; a_0+2d, \\; a_0+3d, \\; \\ldots \\]</p>\n<p>The difference between a term in the sequence and the term $n$ places along is $n \\times d$.</p>\n<h4>a)</h4>\n<p>The difference between the first two terms is $\\var{n[0]*(index[0]+1)} - \\var{n[0]*index[0]} = \\var{n[0]}$.</p>\n<p>So the common difference is $\\var{n[0]}$.</p>\n<h4>b)</h4>\n<p>We're not given any adjacent terms of this sequence, but we are given some terms two palces apart.</p>\n<p>\\begin{align}<br/>2d &amp;= \\var{n[1]*(index[1]+2)} - \\var{n[1]*index[1]} \\\\<br/>&amp;= \\var{2*n[1]} \\\\<br/>d &amp;= \\var{n[1]}<br/>\\end{align}</p>\n<p>The common difference is $\\var{n[1]}$.</p>\n<h4>c)</h4>\n<p>Again we're not given any adjacent terms of this sequence, but we have two terms three places apart.</p>\n<p>\\begin{align}<br/>3d &amp;= \\var{n[2]*(index[2]+3)} - \\var{n[2]*index[2]} \\\\<br/>&amp;= \\var{3*n[2]} \\\\<br/>d &amp;= \\var{n[2]}<br/>\\end{align}</p>\n<p>The common difference is $\\var{n[2]}$.</p>\n<h4></h4>", "rulesets": {}, "extensions": [], "name": "Find common difference in arithmetic sequences with gaps", "variablesTest": {"condition": "", "maxRuns": 100}, "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/"}]}