// Numbas version: exam_results_page_options
{"metadata": {"description": "<p>Questions on arithmetic, or linear, sequences.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "name": "Arithmetic sequences", "type": "exam", "navigation": {"browse": true, "allowregen": true, "preventleave": true, "showfrontpage": true, "onleave": {"action": "none", "message": ""}, "reverse": true, "showresultspage": "oncompletion"}, "showQuestionGroupNames": false, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"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}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "type": "jme", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "answer": "{a1}+(n-1){d}", "showFeedbackIcon": true, "answersimplification": "basic", "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "vsetrangepoints": 5, "expectedvariablenames": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "a1+(small-1)*d", "maxValue": "a1+(small-1)*d", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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}} = $&nbsp;[[0]]</span></span></p>\n<p><em></em></p>"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "a1+(large-1)*d", "maxValue": "a1+(large-1)*d", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "prompt": "<p>Find the&nbsp;<span color=\"#000000\">$\\var{large}^{\\text{th}}$ term</span></p>\n<p><span color=\"#000000\">$a_{\\var{large}} = $[[0]]</span></p>"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.</p>"}, "tags": ["arithmetic sequences", "nth term", "sequences", "taxonomy"], "variables": {"large": {"templateType": "anything", "description": "<p>A large index to compute</p>", "definition": "random(10..50#5)*10", "name": "large", "group": "Ungrouped variables"}, "small": {"templateType": "anything", "description": "<p>A&nbsp;small index to compute</p>", "definition": "random(6..10)", "name": "small", "group": "Ungrouped variables"}, "a1": {"templateType": "anything", "description": "<p>The first&nbsp;term in the sequence</p>", "definition": "random(1..90)", "name": "a1", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "description": "", "definition": "random(3..13 except 10)", "name": "d", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "d", "small", "large"], "statement": "<p><span color=\"#000000\" style=\"color: #000000;\">In this question, consider the sequence</span></p>\n<p><span color=\"#000000\" style=\"color: #000000;\">\\[ a = \\var{a1}, \\; \\var{a1+d}, \\; \\var{a1+d*2}, \\; \\var{a1+d*3}, \\; \\ldots \\]</span></p>\n<p><span color=\"#000000\" style=\"color: #000000;\"></span>A helpful person has drawn out a table of the terms so far.</p>\n<table>\n<tbody>\n<tr>\n<th>$\\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></h4>\n<h4></h4>", "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Fill in the gaps in an arithmetic sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{m[0]*(index[0]+3)}", "maxValue": "{m[0]*(index[0]+3)}", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{m[0]*(index[0]+5)}", "maxValue": "{m[0]*(index[0]+5)}", "marks": 1, "variableReplacements": []}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "<p>$\\var{m[0]*(index[0])}, \\; \\var{m[0]*(index[0]+1)}, \\; \\var{m[0]*(index[0]+2)}, \\;$&nbsp;[[0]] $, \\; \\var{m[0]*(index[0]+4)}, \\;$ [[1]]$, \\; \\var{m[0]*(index[0]+6)}$</p>\n<p><em></em></p>"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{m[1]*(index[1]-2)}", "maxValue": "{m[1]*(index[1]-2)}", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{m[1]*(index[1]-5)}", "maxValue": "{m[1]*(index[1]-5)}", "marks": 1, "variableReplacements": []}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "<p></p>\n<p>$\\var{m[1]*(index[1])}, \\; \\var{m[1]*(index[1]-1)}, \\;$&nbsp;[[0]]$, \\; \\var{m[1]*(index[1]-3)}, \\; \\var{m[1]*(index[1]-4)}, \\;$ [[1]] $, \\; \\var{m[1]*(index[1]-6)}$</p>"}], "advice": "<p>In an arithmetic sequence, the difference between two adjecent&nbsp;terms is always the same.</p>\n<h4>a)</h4>\n<p>The difference between the first two terms in this sequence is $\\var{m[0]*(index[0]+1)} - \\var{m[0]*(index[0])} = \\var{m[0]}$.</p>\n<p>So to get from one term to the next, add&nbsp;$\\var{m[0]}$.</p>\n<p>The complete sequence is&nbsp;</p>\n<p>\\[ \\var{m[0]*(index[0])}, \\; \\var{m[0]*(index[0]+1)}, \\; \\var{m[0]*(index[0]+2)}, \\; \\underline{\\var{m[0]*(index[0]+3)}}, \\; \\var{m[0]*(index[0]+4)}, \\; \\underline{\\var{m[0]*(index[0]+5)}}, \\; \\var{m[0]*(index[0]+6)} \\]</p>\n<h4>b)</h4>\n<p>The difference between the first two terms in this sequence is $\\var{m[1]*(index[1]-1)} - \\var{m[1]*(index[1])} = \\var{-m[1]}$.</p>\n<p>So to get from one term to the next, subtract&nbsp;$\\var{m[1]}$.</p>\n<p>The complete sequence is&nbsp;</p>\n<p>\\[ \\var{m[1]*(index[1])}, \\; \\var{m[1]*(index[1]-1)}, \\; \\underline{\\var{m[1]*(index[1]-2)}}, \\; \\var{m[1]*(index[1]-3)}, \\; \\var{m[1]*(index[1]-4)}, \\; \\underline{\\var{m[1]*(index[1]-5)}}, \\; \\var{m[1]*(index[1]-6)} \\]</p>\n<p><em></em></p>\n<p></p>\n<h4><em></em></h4>", "tags": ["Arithmetic sequences", "arithmetic sequences", "sequences", "taxonomy"], "variables": {"m": {"templateType": "anything", "description": "<p>Common differences of the sequences</p>", "definition": "shuffle(5..12 except 10)[0..2]", "name": "m", "group": "Ungrouped variables"}, "index": {"templateType": "anything", "description": "<p>Starting indices of the sequences.</p>", "definition": "shuffle(7..15 except 10)[0..2]", "name": "index", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["m", "index"], "statement": "<p>Fill in the gaps in the following arithmetic sequences.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given arithmetic sequences with some terms missing, fill in the missing terms.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Find common difference in arithmetic sequences with gaps", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{n[0]}", "maxValue": "{n[0]}", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{n[1]}", "maxValue": "{n[1]}", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{n[2]}", "maxValue": "{n[2]}", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "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>"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given sequences with missing terms, find the common difference between terms.</p>"}, "tags": ["Arithmetic Sequences", "Arithmetic sequences", "arithmetic sequences", "common difference", "sequences", "taxonomy"], "variables": {"n": {"templateType": "anything", "description": "", "definition": "shuffle(3..9)[0..3]", "name": "n", "group": "Ungrouped variables"}, "index": {"templateType": "anything", "description": "", "definition": "shuffle(5..15 except 10)[0..3]", "name": "index", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["n", "index"], "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>", "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Partial sum of an arithmetic sequence - birthday money", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "variable_groups": [{"variables": ["person", "pronouns"], "name": "A person"}], "functions": {}, "rulesets": {}, "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b", "first"], "metadata": {"description": "<p>The amount of money a person gets on their birthday&nbsp;follows an arithmetic sequence.</p>\n<p>Calculate the amount&nbsp;on a given birthday, then calculate the sum up to that point.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "<p>We are told that {person['name']}'s parents&nbsp;deposit a uniformly&nbsp;increasing&nbsp;amount of money into&nbsp;a savings account&nbsp;for {person['name']} every&nbsp;year on&nbsp;{person['name']}'s&nbsp;birthday.</p>\n<p>We are&nbsp;also given the amount of money that&nbsp;{person['pronouns']['their']} parents deposit into the account&nbsp;on&nbsp;{person['pronouns']['their']} first $3$ birthdays:</p>\n<ul>\n<li>On {person['pronouns']['their']}&nbsp;$1^{st}$ birthday, they deposited $&pound;\\var{first}$ into the account.</li>\n<li>On {person['pronouns']['their']}&nbsp;$2^{nd}$ birthday, they deposited $&pound;\\var{b[1]+first}$ into the account.</li>\n<li>On {person['pronouns']['their']}&nbsp;$3^{rd}$ birthday, they deposited $&pound;\\var{b[1]*2+first}$ into the account.</li>\n</ul>\n<h4>a)</h4>\n<p>To calculate the&nbsp;amount of money&nbsp;{person['name']}'s parents&nbsp;would deposit&nbsp;into the savings account on&nbsp;{person['pronouns']['their']} 21<sup>st</sup>&nbsp;birthday, if {pronouns['their']} parents maintained this pattern, we&nbsp;use the equation</p>\n<p>\\[a_n=a_1+(n-1)d\\text{,}\\]</p>\n<p>where</p>\n<ul>\n<li>$a_1$ is the first&nbsp;term;</li>\n<li>$a_n$ is the $n^{\\text{th}}$ term;</li>\n<li>$d$ is the common difference between consecutive terms.</li>\n</ul>\n<p>To identify the first term and common difference of the sequence we can use a table like the one below.</p>\n<table border=\"1\">\n<thead></thead>\n<tbody>\n<tr>\n<th>$n$</th>\n<td>$1$</td>\n<td>$2$</td>\n<td>$3$</td>\n</tr>\n<tr>\n<th>$a_n$</th>\n<td>$\\mathbf{\\var{first}}$</td>\n<td>$\\var{b[1]+first}$</td>\n<td>$\\var{b[1]*2+first}$</td>\n</tr>\n<tr>\n<th>First differences</th>\n<td></td>\n<td>$\\mathbf{\\var{b[1]}}$</td>\n<td>$\\mathbf{\\var{b[1]}}$</td>\n</tr>\n</tbody>\n</table>\n<p>The first term&nbsp;and common difference have been highlighted in bold: $a_1 = \\var{first}$ and $d = \\var{b[1]}$.</p>\n<p>Now we&nbsp;can use these to calculate $a_{21}$, giving us</p>\n<p>\\begin{align}<br/>a_{21}&amp;=\\var{first}+\\var{b[1]} \\times (21-1) \\\\<br/>&amp;=\\var{first+b[1]*(20)}\\text{.} \\\\<br/>\\end{align}</p>\n<p>So, assuming that {person['name']}'s parents do maintain this pattern, on {pronouns['their']}&nbsp;21<sup>st</sup>&nbsp;birthday {pronouns['their']} parents will&nbsp;deposit $&pound;\\var{first+b[1]*(20)}$ into the savings account.</p>\n<h4>b)</h4>\n<p>We are now asked to calculate&nbsp;the total&nbsp;amount of&nbsp;money that {person['name']}'s&nbsp;parents will&nbsp;have added to this savings account over 21 years, including the money that&nbsp;{pronouns['their']} parents will&nbsp;deposit into the account on {pronouns['their']} 21<sup>st</sup>&nbsp;birthday.</p>\n<p>This question involves calculating the sum using the equation</p>\n<p>\\[\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\\]</p>\n<p>We know from part <strong>a) </strong>that</p>\n<p>\\begin{align}<br/>a_1&amp;=\\var{first},\\\\<br/>n&amp;=21,\\\\<br/>a_{21}&amp;= \\var{first+b[1]*(20)}.<br/>\\end{align}</p>\n<p>Using&nbsp;our formula for the sum,</p>\n<p>\\begin{align}<br/>\\sum\\limits_{i=1}^n{a_i}&amp;=\\frac{n}{2}(a_1+a_n)\\\\<br/>&amp;=\\frac{\\var{21}}{2}(\\var{first}+\\var{first+b[1]*(21-1)})\\\\<br/>&amp;=\\var{21*(first+first+b[1]*(20))/2}\\text{.}<br/>\\end{align}</p>\n<p>Therefore, over 21 years {person['name']}'s&nbsp;parents will&nbsp;have added a total of $&pound;\\var{21*(first+first+b[1]*(20))/2}$&nbsp;to this savings account!</p>", "statement": "<p>{person['name']}'s parents&nbsp;deposit a uniformly&nbsp;increasing&nbsp;amount of money into&nbsp;a savings account&nbsp;for {pronouns['them']} every&nbsp;year on {pronouns['their']}&nbsp;birthday:</p>\n<ul>\n<li>on&nbsp;{pronouns['their']} first&nbsp;birthday&nbsp;{pronouns['their']} parents deposited&nbsp;$&pound;\\var{first}$ into the account;</li>\n<li>on&nbsp;{pronouns['their']} second&nbsp;birthday&nbsp;{person['pronouns']['their']} parents&nbsp;deposited&nbsp;$&pound;\\var{b[1]+first}$ into the account;</li>\n<li>on {pronouns['their']} third&nbsp;birthday&nbsp;{pronouns['their']} parents&nbsp;deposited $&pound;\\var{b[1]*2+first}$ into the account.</li>\n</ul>\n<p>{person['name']} wants to know&nbsp;the total amount&nbsp;of&nbsp;money&nbsp;that will be in&nbsp;this savings account,&nbsp;excluding interest,&nbsp;after&nbsp;{pronouns['their']} 21<sup>st</sup>&nbsp;birthday, if&nbsp;{pronouns['their']} parents maintain this pattern.</p>", "preamble": {"js": "", "css": ""}, "variables": {"c": {"name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(3..13 except[10]),8)"}, "n": {"name": "n", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(3..9),7)"}, "person": {"name": "person", "description": "<p>A random person</p>", "templateType": "anything", "group": "A person", "definition": "random_person()"}, "m": {"name": "m", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(2..10),5)"}, "first": {"name": "first", "description": "<p>first term in the sequence</p>", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..15 #5)"}, "pronouns": {"name": "pronouns", "description": "", "templateType": "anything", "group": "A person", "definition": "person['pronouns']"}, "ni": {"name": "ni", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(19..40),10)"}, "b": {"name": "b", "description": "<p>a</p>", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(10..25 #5), 3)"}, "ci": {"name": "ci", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(6..20),10)"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "<p>How much money will&nbsp;{person['name']}'s parents deposit&nbsp;into the savings account on {pronouns['their']} 21<sup>st</sup>&nbsp;birthday,&nbsp;if {pronouns['their']} parents maintain this pattern?&nbsp;</p>\n<p>&pound;[[0]].</p>", "stepsPenalty": 0, "gaps": [{"answer": "{first+b[1]*(20)}", "showpreview": true, "expectedvariablenames": [], "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "type": "jme", "checkingaccuracy": 0.001, "variableReplacements": [], "vsetrange": [0, 1], "checkvariablenames": false, "showFeedbackIcon": true, "scripts": {}, "marks": 1, "showCorrectAnswer": true}], "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "<p>Use the arithmetic formula,</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&nbsp;term in&nbsp;the sequence;</li>\n<li>$d$ is the common difference between consecutive terms.</li>\n</ul>", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{first}", "allowFractions": false, "prompt": "<p>What is the value of $a_1$?</p>", "mustBeReducedPC": 0, "minValue": "{first}", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": "0.2", "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{b[1]}", "allowFractions": false, "prompt": "<p>What is the value of $d$?</p>", "mustBeReducedPC": 0, "minValue": "{b[1]}", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": "0.2", "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "prompt": "<p>Now use the formula to calculate&nbsp;$a_{21}$.</p>", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "<p>How much money will&nbsp;{person['name']}'s&nbsp;parents have added to this savings account over&nbsp;$21$ years in total, including the money that&nbsp;{person['pronouns']['their']} parents will&nbsp;deposit into the account on {person['pronouns']['their']} $21^{st}$ birthday?</p>\n<p>&pound;[[0]].</p>\n<p></p>", "stepsPenalty": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{21*(first+first+b[1]*(20))/2}", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "{21*(first+first+b[1]*(20))/2}", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 1, "scripts": {}}], "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "<p>The sum of an arithmetic sequence $a_1, a_2, \\ldots, a_n$ is calculated by the following formula.</p>\n<p>\\[\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\\]</p>", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "tags": ["Arithmetic sequences", "Arithmetic Sequences", "arithmetic sequences", "nth term", "partial sums", "random names", "sequences", "taxonomy"], "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Arithmetic sequences in an ice cream shop", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": ["index", "d", "person"], "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>", "variable_groups": [], "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>", "parts": [{"correctAnswerFraction": false, "stepsPenalty": 0, "mustBeReducedPC": 0, "prompt": "<p>How many customers before {person['name']} have tried the strawberry ice-cream?</p>", "showFeedbackIcon": true, "allowFractions": false, "minValue": "index", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "index", "mustBeReduced": false, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "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": [], "showFeedbackIcon": true, "marks": 0}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "prompt": "<p>For this arithmetic sequence, what is $a_1$?</p>", "showFeedbackIcon": true, "allowFractions": false, "minValue": "1", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "1", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain"}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "prompt": "<p>What is $d$?</p>", "showFeedbackIcon": true, "allowFractions": false, "minValue": "{d}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{d}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain"}], "marks": "3", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "tags": ["arithmetic sequence", "calculate the term number", "common difference", "nth term", "sequences", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"person": {"description": "", "group": "Ungrouped variables", "definition": "random_person()", "name": "person", "templateType": "anything"}, "index": {"description": "", "group": "Ungrouped variables", "definition": "random(14..34 except [20, 21, 22, 30, 31, 32, 40])", "name": "index", "templateType": "anything"}, "d": {"description": "", "group": "Ungrouped variables", "definition": "random(6..12 except 10)", "name": "d", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "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}, "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/"}], "type": "question", "statement": "<p>A linear 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>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"ni": {"group": "Ungrouped variables", "name": "ni", "description": "", "templateType": "anything", "definition": "repeat(random(19..40),10)"}, "c": {"group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything", "definition": "repeat(random(3..13 except[10]),8)"}, "n": {"group": "Ungrouped variables", "name": "n", "description": "", "templateType": "anything", "definition": "repeat(random(1..4),7)"}, "m": {"group": "Ungrouped variables", "name": "m", "description": "", "templateType": "anything", "definition": "repeat(random(2..10),5)"}, "b": {"group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything", "definition": "repeat(random(2..4), 5)"}, "ci": {"group": "Ungrouped variables", "name": "ci", "description": "", "templateType": "anything", "definition": "repeat(random(6..20),10)"}}, "functions": {}, "tags": ["arithmetic formula", "common difference", "first term", "formula for the nth term", "linear sequences", "nth term", "sequences", "taxonomy"], "variable_groups": [], "parts": [{"variableReplacements": [], "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answer": "{m[1]}*n+{m[1]}", "marks": "3", "variableReplacements": [], "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "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>", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "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>", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[1]*2}", "showFeedbackIcon": true, "prompt": "<p>For this arithmetic sequence, what is $a_1$?</p>", "minValue": "{m[1]*2}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[1]}", "showFeedbackIcon": true, "prompt": "<p>What is $d$?</p>", "minValue": "{m[1]}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "marks": 0}, {"variableReplacements": [], "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answer": "-{m[2]}*n+{m[2]*9+2}", "marks": "3", "variableReplacements": [], "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "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>", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "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>", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[2]*8+2}", "showFeedbackIcon": true, "prompt": "<p>For this arithmetic sequence, what is $a_1$?</p>", "minValue": "{m[2]*8+2}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{-m[2]}", "showFeedbackIcon": true, "prompt": "<p>What is $d$?</p>", "minValue": "{-m[2]}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "marks": 0}], "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "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>"}, {"name": "Compute the partial sum of an arithmetic sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "metadata": {"description": "<p>Given the first and last terms of a finite arithmetic sequence, calculate the number of elements and then the sum of the sequence.</p>\n<p>Each part is broken into steps, with the formula given.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["m", "first_index", "last_index", "num_terms", "first_term", "last_term", "partial_sum"], "type": "question", "rulesets": {}, "advice": "<p>We are given the arithmetic sequence</p>\n<p>\\[\\var{m*first_index}, \\var{m*(first_index+1)}, \\var{m*(first_index+1)}, \\ldots, \\var{last_term} \\text{.} \\]</p>\n<h4>a)</h4>\n<p>Let $n$ be the number of terms in the sequence. Then $a_n = \\var{last_term}$.</p>\n<p>To find $n$, we must rearrange the formula for the $n^\\text{th}$ term</p>\n<p>\\[a_n=a_1+(n-1)d\\text{.}\\]</p>\n<p>The first term is $a_1 = \\var{first_term}$ and the common difference is $d = \\var{first_term+m} - \\var{first_term} = \\var{m}$.</p>\n<p>So we have</p>\n<p>\\begin{align}<br/>d&amp;=\\var{m} \\text{,} \\\\<br/>a_1&amp;= \\var{first_term} \\text{,} \\\\<br/>a_n&amp;=\\var{last_term} \\text{.}<br/>\\end{align}</p>\n<p>We now substitute these values into the formula for the $n^\\text{th}$ term and rearrange to find $n$.</p>\n<p>\\begin{align}<br/>\\var{last_term}&amp;=\\var{first_term}+\\var{m}(n-1) \\\\<br/>\\var{last_term}-\\var{first_term}&amp;=\\var{m}(n-1) \\\\<br/>\\frac{\\var{last_term-first_term}}{\\var{m}}&amp;=(n-1) \\\\<br/>n&amp;=\\var{(last_term-first_term)/m}+1 = \\var{num_terms}\\text{.}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>The partial sum of the first $n$ terms of an arithmetic sequence is given by</p>\n<p>\\[\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\\]</p>\n<p>We know from part <strong>a) </strong>that</p>\n<p>\\begin{align}<br/>n&amp;= \\var{num_terms} \\text{,} \\\\<br/>a_1&amp;= \\var{first_term} \\text{,} \\\\<br/>a_n&amp;= \\var{last_term} \\text{.}<br/>\\end{align}</p>\n<p>We substitute these values into the formula, obtaining</p>\n<p>\\begin{align}<br/>\\sum\\limits_{i=1}^n{a_i}&amp;=\\frac{n}{2}(a_1+a_n)\\\\<br/>&amp;= \\frac{\\var{num_terms}}{2}(\\var{first_term}+\\var{last_term})\\\\<br/>&amp;= \\simplify[]{ ({num_terms}*({first_term+last_term}))/2} \\\\<br/>&amp;= \\var{partial_sum} \\text{.}<br/>\\end{align}</p>", "variables": {"last_index": {"description": "<p>The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.</p>\n<p>This is the index of the last term shown.</p>", "definition": "first_index+num_terms-1", "group": "Ungrouped variables", "name": "last_index", "templateType": "anything"}, "m": {"description": "<p>Common difference between terms</p>", "definition": "random(2..10)", "group": "Ungrouped variables", "name": "m", "templateType": "anything"}, "first_index": {"description": "<p>The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.</p>\n<p>This is the index of the first term shown.</p>", "definition": "random(6..14)", "group": "Ungrouped variables", "name": "first_index", "templateType": "anything"}, "num_terms": {"description": "<p>The number of terms in the sequence.</p>", "definition": "random(11..30)", "group": "Ungrouped variables", "name": "num_terms", "templateType": "anything"}, "partial_sum": {"description": "<p>Sum of the terms in the sequence.</p>", "definition": "num_terms*(first_term+last_term)/2", "group": "Ungrouped variables", "name": "partial_sum", "templateType": "anything"}, "first_term": {"description": "<p>The first term in the sequence.</p>", "definition": "m*first_index", "group": "Ungrouped variables", "name": "first_term", "templateType": "anything"}, "last_term": {"description": "<p>The last term in the sequence.</p>", "definition": "m*last_index", "group": "Ungrouped variables", "name": "last_term", "templateType": "anything"}}, "statement": "<p>You are given the following arithmetic sequence:</p>\n<p>\\[\\var{m*first_index}, \\var{m*(first_index+1)},&nbsp;\\var{m*(first_index+2)}, \\ldots, \\var{last_term}.\\]</p>", "parts": [{"scripts": {}, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>The formula for the $n^\\text{th}$ term in an arithmetic sequence is</p>\n<p>\\[a_n=a_1+(n-1)d\\text{,}\\]</p>\n<p>where</p>\n<ul>\n<li>$d$ is the common difference between consecutive terms;</li>\n<li>$a_1$ is the first term in the sequence;</li>\n<li>$a_n$ is the $n^\\text{th}$ term in the sequence.</li>\n</ul>"}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "prompt": "<p>For this arithmetic sequence, what is the value of $a_1$?</p>", "showFeedbackIcon": true, "allowFractions": false, "minValue": "first_term", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "first_term", "mustBeReduced": false, "marks": "0.2", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "prompt": "<p>What is the value of $d$?</p>", "showFeedbackIcon": true, "allowFractions": false, "minValue": "m", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "m", "mustBeReduced": false, "marks": "0.2", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "num_terms", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "num_terms", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "marks": 0, "prompt": "<p>How many terms are there in this sequence?</p>\n<p>$n =$ [[0]]</p>", "stepsPenalty": 0}, {"scripts": {}, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>The sum of an arithmetic sequence $a_1, a_2, \\ldots, a_n$ is calculated&nbsp;using the formula</p>\n<p>\\[ \\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n) \\]</p>\n<p>where</p>\n<ul>\n<li>$a_1$ is the first term;</li>\n<li>$a_n$ is the $n^{th}$&nbsp;term;</li>\n<li>$n$ is the number of terms.</li>\n</ul>"}], "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "partial_sum", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "partial_sum", "mustBeReduced": false, "marks": "2", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "marks": 0, "prompt": "<p>Find the sum of the sequence up to and including the term $\\var{last_term}$.</p>\n<p>[[0]]</p>", "stepsPenalty": 0}], "tags": ["arithmetic sequences", "Arithmetic sequences", "Arithmetic Sequences", "common difference", "partial sums", "sequences", "taxonomy", "term number"], "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "feedback": {"allowrevealanswer": true, "advicethreshold": 0, "intro": "", "showanswerstate": true, "feedbackmessages": [], "showtotalmark": true, "showactualmark": true}, "timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "showstudentname": true, "percentPass": 0, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": ["random_person"], "custom_part_types": [], "resources": []}