// Numbas version: exam_results_page_options
{"name": "Deirdre's copy of Find and use the formula for a geometric sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "r", "n", "nth_term"], "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "functions": {}, "preamble": {"css": "", "js": ""}, "tags": [], "variable_groups": [], "parts": [{"gaps": [{"correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "r", "variableReplacements": [], "scripts": {}, "mustBeReducedPC": 0, "type": "numberentry", "marks": 1, "maxValue": "r"}], "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "prompt": "<p>Find the common ratio for&nbsp;the following geometric series.</p>\n<p>$\\var{a}, \\var{a*r}, \\var{a*r^2}, \\var{a*r^4}, \\ldots$</p>\n<p>Common ratio:&nbsp;[[0]]<em></em></p>"}, {"gaps": [{"vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "type": "jme", "answer": "{a}*{r}^(n-1)"}], "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "steps": [{"showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "information", "marks": 0, "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>"}], "showFeedbackIcon": true, "stepsPenalty": 0, "type": "gapfill", "marks": 0, "prompt": "<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>"}, {"gaps": [{"correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "a*r^n", "variableReplacements": [], "scripts": {}, "mustBeReducedPC": 0, "type": "numberentry", "marks": 1, "maxValue": "a*r^n"}], "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "prompt": "<p><em></em>What is the $\\var{n}^\\text{th}$ term in this sequence?</p>\n<p>$a_\\var{n} =$&nbsp;[[0]]</p>"}], "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 } \\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 $a_n = \\simplify[]{ {a}*{r}^n}$.&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{b}} \\\\<br/>&amp;= \\var{a*r^n}<br/>\\end{align}</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "rulesets": {}, "variables": {"r": {"templateType": "anything", "description": "<p>The common ratio</p>", "definition": "random(3..8)", "group": "Ungrouped variables", "name": "r"}, "nth_term": {"templateType": "anything", "description": "", "definition": "a*r^n", "group": "Ungrouped variables", "name": "nth_term"}, "n": {"templateType": "anything", "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>", "definition": "random(ceil(log(1000,r)-log(a,r))..floor(log(1000000,r)-log(a,r)))", "group": "Ungrouped variables", "name": "n"}, "a": {"templateType": "anything", "description": "<p>The first term</p>", "definition": "random(3..10 except r)", "group": "Ungrouped variables", "name": "a"}}, "statement": "", "name": "Deirdre's copy of Find and use the formula for a geometric sequence", "type": "question", "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}]}]}], "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}]}