// Numbas version: finer_feedback_settings
{"name": "Arithmetic sequences in an ice cream shop", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given&nbsp;the common difference and first term of an arithmetic sequence, work out the index of the nth term of the sequence.</p>\n<p>Framed as a word problem&nbsp;with ticket numbers in an ice cream shop.</p>"}, "tags": ["arithmetic sequence", "calculate the term number", "common difference", "nth term", "sequences", "taxonomy"], "functions": {}, "name": "Arithmetic sequences in an ice cream shop", "ungrouped_variables": ["index", "d", "person"], "parts": [{"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "steps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "prompt": "<p>Use the arithmetic formula,</p>\n<p>\\[a_n = a_1 + (n-1)d \\]</p>\n<p>where<br/>$a_n$ - The n$^{th}$ term in an arithmetic sequence&nbsp;<br/>$a_1$ - The $1^{st}$ term in an arithmetic sequence&nbsp;<br/>$n$ - Term number<br/>$d$ - The common difference.</p>", "variableReplacements": [], "scripts": {}, "marks": 0}, {"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "1", "scripts": {}, "marks": 1, "allowFractions": false, "type": "numberentry", "prompt": "<p>For this arithmetic sequence, what is $a_1$?</p>", "correctAnswerFraction": false, "maxValue": "1"}, {"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "{d}", "scripts": {}, "marks": 1, "allowFractions": false, "type": "numberentry", "prompt": "<p>What is $d$?</p>", "correctAnswerFraction": false, "maxValue": "{d}"}], "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "index", "scripts": {}, "marks": "3", "allowFractions": false, "stepsPenalty": 0, "type": "numberentry", "prompt": "<p>How many customers before {person['name']} have tried the strawberry ice-cream?</p>", "correctAnswerFraction": false, "maxValue": "index"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"person": {"description": "", "definition": "random_person()", "name": "person", "templateType": "anything", "group": "Ungrouped variables"}, "index": {"description": "", "definition": "random(14..34 except [20, 21, 22, 30, 31, 32, 40])", "name": "index", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"description": "", "definition": "random(6..12 except 10)", "name": "d", "templateType": "anything", "group": "Ungrouped variables"}}, "extensions": ["random_person"], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "<p>We know that every $\\var{d}^{\\text{th}}$ ticket after the first receives strawberry ice cream. So, the sequence of ticket numbers which get strawberry ice cream starts like this:</p>\n<p>\\[ 1, \\var{d+1}, \\var{2d+1}, \\var{3d+1}, \\ldots \\]</p>\n<p>The numbers on the tickets for strawberry ice cream form an arithmetic sequence: the first term is $1$ and the common difference is $\\var{d}$.</p>\n<p>We can write down a formula for the $n^{\\text{th}}$ term in this sequence and rearrange it to find how many customers received strawberry ice cream before {person['name']}.</p>\n<p>The general formula for 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 $n^{th}$ term;</li>\n<li>$a_1$ is the first term;</li>\n<li>$d$ is the common difference between consecutive terms.</li>\n</ul>\n<p>We know that $a_n = \\var{1+d*index}$, $a_1 = 1$, and $d = \\var{d}$, and we want to find $n$.</p>\n<p>Substituting these values from the formula gives</p>\n<p>\\begin{align}<br/>a_n &amp;= a_1 + (n-1)d \\\\<br/>\\var{1+d*index} &amp;= 1 + (n -1)\\var{d}\\,.<br/>\\end{align}</p>\n<p>Now we rearrange this to find $n$:</p>\n<p>\\begin{align}<br/>\\var{1+d*index-1} &amp;= \\var{d}n - \\var{d} \\\\<br/>\\var{d*index +d} &amp;= \\var{d}n \\\\<br/>n &amp;= \\var{(d*index + d)/d}.<br/>\\end{align}</p>\n<p>This is the number of people who received strawberry ice cream&nbsp;up to and&nbsp;including {person['name']}. Removing {person['name']} leaves $\\var{index}$ customers who received strawberry ice cream before {person['name']} did.</p>", "statement": "<p>When customers enter an ice cream shop they receive a numbered ticket for a free sample.</p>\n<p>There are $\\var{d}$ flavours of ice cream&nbsp;that the shop alternates&nbsp;through sequentially.</p>\n<p>The first ticket was number $1$ and the person with this ticket received strawberry ice cream.</p>\n<p>{person['name']} was given ticket number&nbsp;$\\var{1+d*(index)}$ and also received strawberry ice cream.</p>", "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": "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": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}