// Numbas version: exam_results_page_options
{"name": "CF Maths In class test three mock paper question 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["t1", "tr", "tsum", "r"], "name": "CF Maths In class test three mock paper question 3", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Part a)</p>\n<p>$a$ is the first term in the sequence. Namely, $\\var{t1}$</p>\n<p>To find the common ratio, $r$, simply divide one term by its predecessor.</p>\n<p>Part b)</p>\n<p>To find the $i^{th}$<sup>&nbsp;</sup>term, $a_i$, for a specific $i$, recall the sigma notation&nbsp;for the sum of a geometric series:</p>\n<p>$\\displaystyle\\sum\\limits_{i=1}^nar^{i-1}$ $_{..(I)}$ &nbsp; also, note that any individual term is found by: $a_i=ar^{i-1}$&nbsp;<strong></strong>$_{..(II)}$</p>\n<p>To calculate $a_\\var{tr}$, substitute the known values of $a$, $r$ and $i$ into&nbsp;$_{(II)}$ and simplify.</p>\n<p>Part c)</p>\n<p>When using the sigma notation $\\sum$ as in&nbsp;$_{(I)}$, each individual term is often labelled with the letter $i$ and the total number of terms in the series is defined as the value for $n$. It is important to recognise what each letter is referring to in each instance to avoid confusion.</p>\n<p>To find the general $i^{th}$ term, $a_i$, formula of a geometric series, we substitute known values of $a$ and $r$ into&nbsp;$_{(II)}$ to produce the formula in $i$.</p>\n<p>In this case,</p>\n<p>$a_i=\\var{t1}(\\var{r})^{i-1}$</p>\n<p>Part d)</p>\n<p>To find the sum of the first $n$ terms of a geometric series, we turn to the following formula:</p>\n<p>$\\displaystyle\\sum\\limits_{i=1}^nar^{i-1}=a\\left(\\frac{1-r^n}{1-r}\\right)$&nbsp;$_{..(III)}$</p>\n<p>The sum is found by subsituting the known values for $a$, $r$ and $n$ into&nbsp;$_{(III)}$.</p>\n<p></p>", "rulesets": {}, "parts": [{"prompt": "<p>State the $a$ and $r$</p>\n<p>$a$=[[1]]</p>\n<p>$r$=[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{r}", "minValue": "{r}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{t1}", "minValue": "{t1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Find the $\\var{tr}$<sup>th&nbsp;</sup>term</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "{t1}*(({r})^{{tr}-1})", "minValue": "{t1}*(({r})^{{tr}-1})", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "prompt": "<p>Find the $i^{th}$&nbsp;term</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "!basic", "scripts": {}, "answer": "{t1}*(({r})^(i-1))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Find the sum of the first $\\var{tsum}$ terms</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{t1}(1-({r})^{tsum})/((1-{r}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "extensions": [], "statement": "<p>For the geometric series:</p>\n<p>$\\var{t1}$<span data-jme-visible=\"{r}&gt;0\">+</span>$\\simplify{{t1}*{r}}$+$\\simplify{{t1}*{{r}^2}}$<span data-jme-visible=\"{r}&gt;0\">+</span>$\\simplify{{t1}*{{r}^3}}$+...</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"tsum": {"definition": "random(4..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "tsum", "description": ""}, "r": {"definition": "random(-4..4 except [1,0,-1])", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "tr": {"definition": "random(4..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "tr", "description": ""}, "t1": {"definition": "tr", "templateType": "anything", "group": "Ungrouped variables", "name": "t1", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}