// Numbas version: finer_feedback_settings
{"duration": 0, "question_groups": [{"name": "Group", "pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "Find and use the formula for a geometric 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": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "tags": ["common ratio", "geometric sequences", "nth term", "sequences", "taxonomy"], "metadata": {"description": "<p>Find the common ratio of a given geometric sequence, write down&nbsp;the formula for the nth term and use it to calculate a given term in the sequence.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>The terms in a geometric sequence are found by repeatedly multiplying the last term by a constant, called the common ratio.</p>\n<h4>a)</h4>\n<p>To find the common ratio, pick a term of the sequence and divide it by the previous term.</p>\n<p>We can calculate the common ratio using a table:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$n$</td>\n<td>$1$</td>\n<td>$2$</td>\n<td>$3$</td>\n<td>$4$</td>\n</tr>\n<tr>\n<td>$a_n$</td>\n<td>$\\var{a}$</td>\n<td>$\\var{a*r}$</td>\n<td>$\\var{a*r^2}$</td>\n<td>$\\var{a*r^3}$</td>\n</tr>\n<tr>\n<td>$a_n \\div a_{n-1}$</td>\n<td></td>\n<td>$\\var{r}$</td>\n<td>$\\var{r}$</td>\n<td>$\\var{r}$</td>\n</tr>\n</tbody>\n</table>\n<p>The common ratio is $\\var{r}$.</p>\n<h4>b)</h4>\n<p>The general formula for the $n^\\text{th}$ term of a geometric sequence is</p>\n<p>\\[\\displaystyle {a_n=ar^{(n-1)}\\text{,}}\\]</p>\n<p>where&nbsp;$a$ is the&nbsp;first term, and $r$ is the common ratio.</p>\n<p>So the formula for this sequence is</p>\n<p>\\[ a_n = \\simplify[]{ {a}*{r}^(n-1) } \\text{.} \\]</p>\n<h4>c)</h4>\n<p>We know from&nbsp;part <strong>b)</strong> that the formula for the&nbsp;$n^\\text{th}$ term is $a_n = \\simplify[]{ {a}*{r}^{(n-1)}}$.&nbsp;</p>\n<p>Therefore&nbsp;the $\\var{n}^\\text{th}$ term in the sequence is</p>\n<p>\\begin{align}<br/>a_\\var{n} &amp;= \\var{a} \\times \\var{r}^{\\var{n-1}} \\\\<br/>&amp;= \\var{a*r^(n-1)}<br/>\\end{align}</p>", "rulesets": {}, "variables": {"nth_term": {"name": "nth_term", "group": "Ungrouped variables", "definition": "a*r^n", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3..10 except r)", "description": "<p>The first term</p>", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "<p>The common ratio</p>", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(ceil(log(1000,r)-log(a,r))..floor(log(1000000,r)-log(a,r)))", "description": "<p>The index of a term to calculate.</p>\n<p>The range is picked so that the number is between 1,000 and 1,000,000.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "r", "n", "nth_term"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the common ratio for&nbsp;the following geometric series.</p>\n<p>$\\var{a}, \\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\ldots$</p>\n<p>Common ratio:&nbsp;[[0]]<em></em></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "r", "maxValue": "r", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The first term in the sequence is {a}.</p>\n<p><em></em>Write down the formula for the &nbsp;$n^\\text{th}$ term in the sequence</p>\n<p>$a_n = $&nbsp;[[0]]<em></em></p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The formula for the $n^\\text{th}$ term of a geometric sequence is</p>\n<p>\\[ a_n = ar^{(n-1)} \\]</p>\n<p>where $a$ is the first&nbsp;term in the sequence and $r$ is the common ratio.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}*{r}^(n-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em></em>What is the $\\var{n}^\\text{th}$ term in this sequence?</p>\n<p>$a_\\var{n} =$&nbsp;[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a*r^(n-1)", "maxValue": "a*r^(n-1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Geometric Sequence - negative ratio ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "advice": "<p>The terms in a geometric sequence are found by repeatedly multiplying the last term by a constant, called the common ratio.</p>\n<h4>a)</h4>\n<p>To find the common ratio, pick a term of the sequence and divide it by the previous term.</p>\n<p>We can calculate the common ratio using a table:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$n$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">$2$</td>\n<td style=\"text-align: center;\">$3$</td>\n<td style=\"text-align: center;\">$4$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$a_n$</td>\n<td style=\"text-align: center;\">$\\var{a*r}$</td>\n<td style=\"text-align: center;\">$\\var{a*r^2}$</td>\n<td style=\"text-align: center;\">$\\var{a*r^3}$</td>\n<td style=\"text-align: center;\">$\\var{a*r^4}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">Common ratio</td>\n<td></td>\n<td style=\"text-align: center;\">$\\displaystyle\\frac{\\var{a*r^2}}{(\\var{a*r})} = \\var{r}$</td>\n<td style=\"text-align: center;\">$\\displaystyle\\frac{\\var{a*r^3}}{\\var{a*r^2}} = \\var{r}$</td>\n<td style=\"text-align: center;\">$\\displaystyle\\frac{\\var{a*r^4}}{(\\var{a*r^3})} = \\var{r}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>The common ratio is $\\var{d}$.</p>\n<h4>b)</h4>\n<p>The general formula for the $n^\\text{th}$ term of a geometric sequence is</p>\n<p>\\[\\displaystyle {a_n=ar^{(n-1)}\\text{,}}\\]</p>\n<p>where&nbsp;$a$ is the&nbsp;first term, and $r$ is the common ratio.</p>\n<p>So the formula for this sequence is</p>\n<p>\\[<br/>\\begin{align}<br/>a_n&amp;=ar^{(n-1)}\\\\<br/>&amp;=\\var{a*r}\\times(\\var{r})^{(n-1)}\\\\<br/>&amp;=(\\var{a} \\times (\\var{r}))(\\var{r})^{n-1}\\\\<br/>&amp;=\\var{a}(\\var{r})^n\\text{.}<br/>\\end{align}<br/>\\]</p>\n<h4>c)</h4>\n<p>We know from part <strong>b)&nbsp;</strong>that the $n^{th}$ term for this sequence is $a_n = \\var{a}(\\var{r})^n$.</p>\n<p>Therefore the $\\var{nth}^{th}$ term in the sequence is&nbsp;</p>\n<p>\\[<br/>\\begin{align}<br/>a_\\var{nth} &amp;= \\var{a}(\\var{r})^\\var{nth}\\\\<br/>&amp;= \\var{a} \\times (\\var{{r}^{nth}})\\\\<br/>&amp;= \\var{{a}*{r}^{nth}}.<br/>\\end{align}<br/>\\]</p>\n<p></p>\n<p></p>", "statement": "", "variables": {"nth": {"name": "nth", "group": "Ungrouped variables", "definition": "random(6,7)", "templateType": "anything", "description": ""}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2,3)", "templateType": "anything", "description": ""}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "-random(3..5)", "templateType": "anything", "description": ""}}, "tags": ["common ratio", "geometric sequences", "negative common ratio", "nth term", "sequences", "taxonomy"], "ungrouped_variables": ["nth", "r", "a"], "functions": {}, "preamble": {"js": "", "css": ""}, "type": "question", "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "r", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "r", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>Find the common ratio for&nbsp;the following geometric series.</p>\n<p>$\\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\var{a*r^4}\\ldots$</p>\n<p>Common ratio =&nbsp;[[0]]</p>\n<p><em></em></p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>The formula for the $n^{th}$ term of a geometric sequence is</p>\n<p>\\[\\displaystyle{ar^{(n-1)}}\\]</p>\n<p>where $a$ is the first term in the sequence and $r$ is the common ratio.</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "gaps": [{"showpreview": true, "checkingtype": "absdiff", "variableReplacements": [], "checkvariablenames": false, "answer": "{a}*{r}^n", "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true, "type": "jme", "marks": 1, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "showFeedbackIcon": true}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Find the formula for the &nbsp;$n^{th}$ term in the sequence:</p>\n<p>$n^{th}$ term =&nbsp;[[0]]<em></em></p>", "marks": 0, "stepsPenalty": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{a}*{r}^{nth}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{a}*{r}^{nth}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>Find the value of the $\\var{nth}^{th}$ term.</p>\n<p>$a_\\var{nth}$ =&nbsp;[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Given a geometric sequence, find the common ratio (negative in this question), write down the formula&nbsp;for the nth term and use it to calculate a given term.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}}]}], "showstudentname": true, "feedback": {"advicethreshold": 0, "intro": "", "showactualmark": true, "feedbackmessages": [], "allowrevealanswer": true, "showtotalmark": true, "showanswerstate": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "navigation": {"preventleave": true, "showresultspage": "oncompletion", "showfrontpage": true, "reverse": true, "allowregen": true, "browse": true, "onleave": {"message": "", "action": "none"}}, "timing": {"timedwarning": 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