// Numbas version: exam_results_page_options
{"name": "Sequences 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["td", "t1", "tr", "tsum", "r1", "r2", "r3", "tans"], "name": "Sequences 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>In this question we're asked to find the&nbsp;value of the term $a_{\\var{r3}}$ (when $i={\\var{r3}}$).</p>\n<p>Given the sum definition for this geometric series provided in the question statement, the formula for the $i^{th}$&nbsp;term&nbsp;would be:</p>\n<p>$a_i=a_0r^i$ $_{(..II)}$ &nbsp;where $a_0$ is the first term of the series and $r$ is the common ratio.</p>\n<p>Substituting the given $i$ we have:</p>\n<p>$a_\\var{r3}=a_0r^{\\var{r3}}$&nbsp;$_{(..III)}$</p>\n<p>Before evaluating this, however, we need to find the values of $r$ and $a_0$.</p>\n<p></p>\n<p><strong>Finding $r$:</strong></p>\n<p>To find the common ratio $r$, we can first divide term $a_\\var{r2}$ by term $a_\\var{r1}$.</p>\n<p>$\\left(\\frac{\\simplify{{t1}/{td}^{r2}}}{\\simplify{{t1}/{td}^{r1}}}\\right)=\\simplify{{{t1}*{td}^{r1}}/{{t1}*{td}^{r2}}}$</p>\n<p>Algebraically speaking, this is the same as:</p>\n<p>$\\left(\\frac{a_0r^{\\var{r2}}}{a_0r^{\\var{r1}}}\\right)=r^{\\simplify{{r2}-{r1}}}$</p>\n<p>$\\therefore\\;\\;\\;r^{\\simplify{{r2}-{r1}}}=\\simplify{{{t1}*{td}^{r1}}/{{t1}*{td}^{r2}}}$</p>\n<p><span data-jme-visible=\"(r2-r1)&gt;1\">$r$ is then found by taking the&nbsp;appropriate root:</span></p>\n<p><span data-jme-visible=\"(r2-r1)&gt;1\">$r=\\sqrt[\\simplify{{r2}-{r1}}]{\\simplify{{{t1}*{td}^{r1}}/{{t1}*{td}^{r2}}}}=\\simplify{1/{td}}$</span></p>\n<p><span data-jme-visible=\"(({t1}*{td}^{r1})/({t1}*{td}^{r2}))=(1/{td})\">Which leaves,</span></p>\n<p><span data-jme-visible=\"(({t1}*{td}^{r1})/({t1}*{td}^{r2}))=(1/{td})\">$r=\\simplify{1/{td}}$</span></p>\n<p><strong></strong></p>\n<p><strong>Finding $a_0$:</strong></p>\n<p>The formula&nbsp;$_{(II)}$ is first rearranged in terms of $a_0$:</p>\n<p>$a_i=a_0r^{i}$ &nbsp; &nbsp;$\\therefore\\;\\;\\;a_0=\\left(\\frac{a_i}{r^{i}}\\right)$</p>\n<p>We can then substitute in the value of $r$, as well as either of the terms provided in the question statement.</p>\n<p>For example, for $a_i=a_\\var{r1}=\\simplify{{t1}/{td}^{r1}}$:</p>\n<p>$a_0=\\left(\\frac{\\simplify{{t1}/{td}^{r1}}}{\\left(\\simplify{1/{td}}\\right)^{\\var{r1}}}\\right)=\\left(\\frac{\\simplify{{t1}/{td}^{r1}}}{\\left(\\simplify{1/{td}}\\right)^{\\var{r1}}}\\right)=\\simplify{{t1}/{td}^{r1}}\\times\\var{td}^{\\var{r1}}=\\var{t1}$</p>\n<p></p>\n<p><b>Finally, for&nbsp;$a_\\var{r3}$:</b></p>\n<p>These values can now be substituted into&nbsp;$_{(III)}$ and simplified:</p>\n<p>$a_\\var{r3}=\\var{t1}(\\simplify{1/{td}})^{\\var{r3}}=\\simplify{{t1}/{{td}^{r3}}}$</p>", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "<p>Assuming the common ratio $r$ is positive, what&nbsp;is the value of the term $a_{\\var{r3}}$ (when $i={\\var{r3}}$)?</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{t1}/{td}^{r3}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "extensions": [], "statement": "<p>We are given&nbsp;a geometric series whose sum is defined by:</p>\n<p>$\\displaystyle\\sum\\limits_{i=0}^na_0r^i=\\displaystyle\\sum\\limits_{i=0}^na_i$&nbsp;$_{..(I)}$</p>\n<p>When $i=\\var{r1}$, the term is calculated to be:&nbsp;$a_{\\var{r1}}=\\simplify{{t1}/{td}^{r1}}$</p>\n<p>Similarly, when&nbsp;$i=\\var{r2}$, the term is calculated to be: $a_{\\var{r2}}=\\simplify{{t1}/{td}^{r2}}$</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "{r2}>{r1}"}, "type": "question", "variables": {"tsum": {"definition": "random(11..29 except 21 except 22 except 23)", "templateType": "anything", "group": "Ungrouped variables", "name": "tsum", "description": ""}, "r1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "r2": {"definition": "random(2..5 except r1)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "r3": {"definition": "r1+r2", "templateType": "anything", "group": "Ungrouped variables", "name": "r3", "description": ""}, "tr": {"definition": "random(4..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "tr", "description": ""}, "t1": {"definition": "(td)^5*random(10..30)", "templateType": "anything", "group": "Ungrouped variables", "name": "t1", "description": ""}, "td": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "td", "description": ""}, "tans": {"definition": "t1/td^r3", "templateType": "anything", "group": "Ungrouped variables", "name": "tans", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}]}]}], "contributors": [{"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}]}