// Numbas version: exam_results_page_options
{"name": "Simon's copy of Finding the formula for the $n^{\\text{th}}$ term of  linear sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Given the first three terms of a sequence, give a formula for the $n^\\text{th}$ term.</p>\n<p>In the first sequence, $d$ is positive. In the second sequence, $d$ is negative.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "advice": "<p>Both of these sequences are linear, or arithmetic, sequences. To find formulas for these sequences we need to identify their&nbsp;<strong>first terms</strong> and&nbsp;<strong>common differences</strong>.</p>\n<h4>a)</h4>\n<p>The formula for the $n^\\text{th}$ term of an arithmetic sequence is</p>\n<p>\\[ a_n = a_1 + (n-1)d \\text{.} \\]</p>\n<p>$a_1$ is the first term and $d$ the common difference between consecutive terms, which we need to identify.</p>\n<p>We can find these by drawing up a table of $a_n$ against $n$, and the differences between consecutive terms.</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;\">First differences</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>The first term&nbsp;and common difference have been highlighted in bold; we can use these to write the formula&nbsp;for the sequence.</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>Similar to the part <strong>a)</strong>, we can identify $a_1$ and $d$ for this sequence by drawing a table of $a_n$ against $n$.</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>$n$</td>\n<td>1</td>\n<td>2</td>\n<td>3</td>\n</tr>\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>First differences</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>The first&nbsp;term&nbsp;and common difference have been&nbsp;highlighted in bold; we can use these to form the&nbsp;formula&nbsp;for the sequence.&nbsp;</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>", "tags": [], "functions": {}, "parts": [{"sortAnswers": false, "stepsPenalty": 0, "marks": 0, "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "gaps": [{"failureRate": 1, "vsetRange": [0, 1], "marks": "3", "checkVariableNames": false, "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "checkingType": "absdiff", "vsetRangePoints": 5, "expectedVariableNames": [], "type": "jme", "variableReplacementStrategy": "originalfirst", "showPreview": true, "showFeedbackIcon": true, "scripts": {}, "answer": "{m[1]}*n+{m[1]}", "checkingAccuracy": 0.001, "customMarkingAlgorithm": ""}], "type": "gapfill", "prompt": "<p>$\\var{m[1]*2}, \\var{m[1]*3},&nbsp;\\var{m[1]*4}, \\ldots$</p>\n<p>$n^\\text{th}$ term =&nbsp;[[0]]</p>", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "scripts": {}, "steps": [{"prompt": "<p>The formula for the $n^{\\text{th}}$ term of an arithmetic sequence is</p>\n<p>\\[a_n = a_1 + (n-1)d, \\]</p>\n<p>where</p>\n<ul>\n<li>$a_n$ is the $n^\\text{th}$ term;</li>\n<li>$a_1$ is the first term in the sequence;</li>\n<li>$d$ is the common difference between consecutive terms.</li>\n</ul>", "type": "information", "variableReplacementStrategy": 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""}], "type": "gapfill", "prompt": "<p>$\\var{m[2]*8+2}, \\var{m[2]*7+2},&nbsp;\\var{m[2]*6+2}...$</p>\n<p>$n^\\text{th}$ term =&nbsp;[[0]]</p>", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "scripts": {}, "steps": [{"prompt": "<p>The formula for the $n^{\\text{th}}$ term of an arithmetic sequence is</p>\n<p>\\[a_n = a_1 + (n-1)d, \\]</p>\n<p>where</p>\n<ul>\n<li>$a_n$ is the $n^\\text{th}$ term;</li>\n<li>$a_1$ is the first term in the sequence;</li>\n<li>$d$ is the common difference between consecutive terms.</li>\n</ul>", "type": "information", "variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true}, {"notationStyles": ["plain", "en", "si-en"], "marks": 1, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, 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"ni", "b"], "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"n": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(1..4),7)", "name": "n"}, "ni": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(19..40),10)", "name": "ni"}, "ci": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(6..20),10)", "name": "ci"}, "c": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(3..13 except[10]),8)", "name": "c"}, "m": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(2..10),5)", "name": "m"}, "b": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(2..4), 5)", "name": "b"}}, "name": "Simon's copy of Finding the formula for the $n^{\\text{th}}$ term of  linear sequences", "variable_groups": [], "statement": "<p>A linear (or arithmetic) sequence is a series of numbers that either increases or decreases by a constant amount at each step.</p>\n<p>Find formulas for the $n^{\\text{th}}$ term for each of the following linear sequences, where the values for $n=1\\text{,}2\\text{,}3$ are given:</p>", "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": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "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": "Simon Thomas", "profile_url": 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