// Numbas version: exam_results_page_options
{"name": "Maths Support: Sequences and limits", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>3 questions. One question on limits of standard sequences. Other two on finding least $N$ such that $|a_n-L |\\lt 10^{-r},\\;\\;n \\geq N$ where $L$ is limit of $(a_n)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 50.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Sequences and Limits 1.1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["ab", "valexp", "v35", "al", "v15", "v150", "v610", "b1", "ga", "b3", "d2", "d1", "val", "s1", "v110", "v560", "s5", "v650", "k1", "r", "v45", "tol", "v210", "v250", "be", "v25", "v65", "de", "v450", "a3", "c1", "d", "a", "c", "b", "v550", "v350", "v510", "k", "v410", "n", "v55", "v310", "v660"], "tags": ["examples of standard limits", "limit", "limits", "limits of sequences", "sequences", "standard limits", "taking the limit"], "preamble": {"css": "", "js": ""}, "advice": "<p>All calculations below are to $5$ decimal places.</p>\n<p>The notation $a \\approx b$ means that $a$ and $b$ are approximately equal.</p>\n<h4>a)</h4>\n<p>Using a calculator for $3$ values of $n$:</p>\n<table border=\"2\">\n  <thead>\n    <tr><th>$n$</th><th>$\\displaystyle{\\frac{1}{n^{1/\\var{r}}}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v15}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v110}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v150}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^{1/\\var{r}}}\\right)=0}$</p>\n<p>In fact $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^r}\\right)=0}$ for any $r \\gt 0$</p>\n<h4>b)</h4>\n<table border=\"2\">\n  <thead>\n    <tr><th>$n$</th><th>$\\displaystyle{\\var{k1}^{1/n}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v25}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v210}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v250}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\var{k1}^{1/n}=1}$, see next question as well.</p>\n<h4>c)</h4>\n<table border=\"2\">\n  <thead>   \n    <tr><th>$n$</th><th>$\\displaystyle{\\var{k}^{1/n}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v35}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v310}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v350}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\var{k}^{1/n}=1}$.</p>\n<p>From the last two questions it seems that $\\displaystyle{\\lim_{n \\to \\infty} k^{1/n}=1}$ for any $k \\gt 0$ &ndash; and this is fact true.</p>\n<h4>d)</h4>\n<table border=\"2\">\n<thead>\n  <tr><th>$n$</th><th>$\\displaystyle{\\frac{\\var{c}n+\\var{d}}{\\var{al}n-\\var{ga}}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v45}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v410}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v450}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that&nbsp; $\\displaystyle \\lim_{n \\to \\infty}\\left(\\simplify[std]{({c}n+{d})/({al}n-{ga})}\\right)\\;=\\;&nbsp;\\simplify[std]{{c}/{al}}$.</p>\n<p>In general <br/>\\[\\lim_{n \\to \\infty}\\left(\\frac{an+b}{cn+d}\\right)= \\frac{a}{c}\\] when $c \\neq 0$</p>\n<h4>e)</h4>\n<table border=\"2\">\n  <thead>\n    <tr><th>$n$</th><th>$\\displaystyle{\\left(\\simplify{{c}/{n}}\\right)^n}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$10$</td><td>$\\var{v55}$</td></tr>\n    <tr><td>$29$</td><td>$\\var{v510}$</td></tr>\n    <tr><td>$50$</td><td>$\\var{v550}$</td></tr>\n    <tr><td>$89$</td><td>$\\var{v560}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify{{c}/{n}}\\right)^n}= 0$. In general $\\displaystyle{\\lim_{n \\to \\infty} r^n= 0}$ if $|r| \\lt 1$</p>\n<h4>f)</h4>\n<p>We have the limit:<br/>\\[\\lim_{n\\to\\infty}\\left(1+\\frac{a}{n}\\right)=e^a\\]<br/>The following table confirms that the values are converging to (five decimal places) $\\displaystyle{\\simplify[std]{e^({a3}/{b3})={valexp}}}$</p>\n<table border=\"2\">\n<thead>\n  <tr><th>$n$</th><th>$\\displaystyle{\\left(\\simplify[std]{1+{a3}/({b3}n)}\\right)^n}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$10$</td><td>$\\var{v65}$</td></tr>\n    <tr><td>$100$</td><td>$\\var{v610}$</td></tr>\n    <tr><td>$1000$</td><td>$\\var{v650}$</td></tr>\n    <tr><td>$10000$</td><td>$\\var{v660}$</td></tr>\n  </tbody>\n</table>\n<p>Hence the answer asked for is $\\var{val}$ to $4$ decimal places.</p>\n<h4>g)</h4>\n<p>The answer to this question is based upon neglecting terms in polynomials in $n$ for large $n$.</p>\n<p>For example, $n^3+1000000n^2+1000000000 \\approx n^3$ for large $n$ as the $n^3$ term completely dominates the other terms as $n \\longrightarrow \\infty$.</p>\n<p>A more precise way of saying this is:<br/>\\[\\lim_{n\\to\\infty}\\left(\\frac{n^3+1000000n^2+1000000000}{n^3}\\right)=1\\]</p>\n<p>So for large $n$<br/>\\[\\begin{eqnarray*} \\frac{\\left(\\simplify[std]{{al^d}n^({a*d})+{be}n^{b}+{c}}\\right)^{1/\\var{d}}} {\\left(\\simplify[std]{{ga^d1}n^({a*d1})+{de}n^{b1}+{c1}}\\right)^{1/\\var{d1}}}&amp;\\approx&amp; \\frac{\\left(\\simplify[std]{{al^d}n^({a*d})}\\right)^{1/\\var{d}}} {\\left(\\simplify[std]{{ga^d1}n^({a*d1})}\\right)^{1/\\var{d1}}}\\\\ &amp;=&amp;\\frac{\\simplify[std]{{al^d}^(1/{d})n^{a}}} {\\simplify[std]{{ga^d1}^(1/{d1})n^{a}}}\\\\ &amp;=&amp;\\simplify[std]{{al}/{ga}} \\end{eqnarray*} \\]<br/>Hence the limit is $\\displaystyle{\\simplify[std]{{al}/{ga}}}$</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^{1/\\var{r}}}\\right)=\\;\\;}$[[0]]</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\var{k1}^{1/n}\\right)=\\;\\;}$[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\var{k}^{1/n}\\right)=\\;\\;}$[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>$\\displaystyle \\lim_{n \\to \\infty}\\left(\\simplify[std]{({c}n+{d})/({al}n-{ga})}\\right)\\;=\\;$[[0]]</p>\n      <p>Enter your answer as a fraction or integer, not as a decimal.</p>\n        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter your answer as a fraction or integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{c}/{al}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify[std]{{c}/{n}}\\right)^n=\\;\\;}$[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify[std]{1+{a3}/({b3}n)}\\right)^n=\\;\\;}$[[0]]<br/>Input your answer to 4 decimal places.</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "val+tol", "minValue": "val-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\frac{\\left(\\simplify[std]{{al^d}n^({a*d})+{be}n^{b}+{c}}\\right)^{1/\\var{d}}}\n      \n      {\\left(\\simplify[std]{{ga^d1}n^({a*d1})+{de}n^{b1}+{c1}}\\right)^{1/\\var{d1}}}=\\;\\;}$[[0]]</p>\n      \n      \n      \n      <p>Enter your answer as a fraction or integer, not as a decimal.</p>\n      \n      \n        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter your answer as a fraction or integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{al}/{ga}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>What are the following limits?</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ab": {"definition": "abs(a3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ab", "description": ""}, "valexp": {"definition": "precround(exp(a3/b3),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "valexp", "description": ""}, "v35": {"definition": "precround(k^(1/100),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v35", "description": ""}, "al": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "al", "description": ""}, "v15": {"definition": "precround(100^(-1/r),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v15", "description": ""}, "v150": {"definition": "precround(5000000^(-1/r),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v150", "description": ""}, "v610": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 100))) ^ 100,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v610", "description": ""}, "b1": {"definition": "a*d1-1", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "ga": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "ga", "description": ""}, "b3": {"definition": "switch(ab=1,random(2..9),ab=2,random(3,5,7,9),ab=3,random(2,4,5,7,8),random(3,5,7,9))", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d2": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d1": {"definition": "if(d2=d,d+1,d2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "val": {"definition": "precround(exp(a3/b3),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "val", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "v110": {"definition": "precround(5000^(-1/r),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v110", "description": ""}, "v560": {"definition": "precround((c/n)^89,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v560", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "v650": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 1000))) ^ 1000,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v650", "description": ""}, "k1": {"definition": "random(100000..200000)", "templateType": "anything", "group": "Ungrouped variables", "name": "k1", "description": ""}, "v55": {"definition": "precround((c/n)^10,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v55", "description": ""}, "v45": {"definition": "precround((c * 100 + d) / (al * 100  -ga),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v45", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "v210": {"definition": "precround(k1^(1/5000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v210", "description": ""}, "v250": {"definition": "precround(k1^(1/5000000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v250", "description": ""}, "be": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "be", "description": ""}, "v25": {"definition": "precround(k1^(1/100),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v25", "description": ""}, "v65": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 10))) ^ 10,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v65", "description": ""}, "de": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "de", "description": ""}, "v450": {"definition": "precround((c * 5000000 + d) / (al * 5000000  -ga),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v450", "description": ""}, "a3": {"definition": "s5*random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "c1": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "v350": {"definition": "precround(k^(1/5000000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v350", "description": ""}, "a": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s1*random(11..50)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "a*d-random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "v550": {"definition": "precround((c/n)^50,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v550", "description": ""}, "d": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "v510": {"definition": "precround((c/n)^29,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v510", "description": ""}, "k": {"definition": "random(2..20#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "description": ""}, "v410": {"definition": "precround((c * 5000 + d) / (al * 5000  -ga),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v410", "description": ""}, "n": {"definition": "abs(c)+random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "r": {"definition": "random(2..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "v310": {"definition": "precround(k^(1/5000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v310", "description": ""}, "v660": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 10000))) ^ 10000,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v660", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>4/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Improved display of prompt for fourth part.</p>\n        \t\t<p>Improved display of solution to fourth part.</p>\n        \t\t<p>Checked calculations.</p>\n        \t\t<p>No tolerance on answer to 6th part, got to be exact to 4dps. Tolerance variable, tol=0.</p>\n        \t\t<p><strong>21/07/2012:</strong></p>\n        \t\t<p>Added description.</p>\n        \t\t<p><strong>27/7/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Question appears to be working correctly.</p>\n        \t\t", "description": "<p>Seven standard elementary limits of sequences.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Sequences and Limits 1.2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {"chcop": {"definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "type": "number", "language": "jme", "parameters": [["a", "number"], ["b", "number"]]}}, "ungrouped_variables": ["a", "c", "b", "d", "n3", "s3", "s2", "s1", "tval", "s5", "s4", "s", "r", "b1", "n1", "n2", "n", "n4", "ep", "t"], "tags": ["", "limit", "limit of sequences", "limits", "sequences", "taking the limit"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>The limit is $\\displaystyle \\simplify[std]{{a}/{c}}$.</p>\n<h4>b)</h4>\n<p>To find the least $N$ such that all terms from the the $N$th are within $10^{\\var{-r}}$ of the limit we proceed as follows:<br/>\\[\\begin{eqnarray*} \\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\leq 10 ^ { -\\var{r}} &amp;\\Leftrightarrow&amp;\\left|\\simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}}\\right| \\leq 10 ^ { -\\var{r}}\\\\ &amp;\\Leftrightarrow&amp;\\simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\\leq 10 ^ { -\\var{r}} \\end{eqnarray*} \\]<br/>(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n+{c*d}} \\gt 0,\\;\\;\\forall n=1,\\;2,\\;3\\ldots$)</p>\n<p>Rearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n+{c*d}})10^{\\var{r}}$ (this is positive and so the inequality does not reverse) we get:<br/>\\[\\simplify[std]{{c^2}n+{c*d}} \\geq \\var{10^r*abs(b*c-a*d)} \\Leftrightarrow n \\geq \\simplify[std]{{1}/{c^2}({10^r*abs(b*c-a*d)}-{c*d})}=\\var{tval}\\]</p>\n<p>Hence the least integer value is given by rounding up i.e. $N=\\var{N}$.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <p>What is the limit of this sequence?</p>\n      \n      \n      \n      <p>$\\displaystyle{\\lim_{x\\to\\infty} x_n=\\;\\;}$[[0]]</p>\n      \n      \n      \n      <p>Input the limit as a fraction or an integer and not a decimal.</p>\n      \n      \n        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter your answer as a fraction or integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a}/{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>Which of the following integers has the property that it is the least integer $N$  such that all  terms in the sequence are within $10^{\\var{-r}}$ of the limit for all $n \\geq N $?</p>\n      \n      \n      \n      <p>[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"displayColumns": 0, "matrix": [1, 0, 0, 0, 0], "shuffleChoices": true, "maxMarks": 0, "distractors": ["", "", "", "", ""], "choices": ["<p>{N}</p>", "<p>{N1}</p>", "<p>{N2}</p>", "<p>{N3}</p>", "<p>{N4}</p>"], "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "minMarks": 0}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Let \\[x_n=\\simplify[std]{({a}n+{b})/({c}n+{d})},\\;\\;n=1,\\;2,\\;3\\ldots\\]</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "chcop(a,a)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "if(a*d=b1*c,b1+1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "chcop(c,c)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "r": {"definition": "random(2,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "b1": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "n": {"definition": "ceil((abs(b*c-a*d)-d*c*ep)/(ep*c^2))", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "t": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "n1": {"definition": "if(N>100,N+s2*random(8..52#4),if(N>50,N+s2*random(8..48#4),N+random(1,3)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "if(N>100,N+s3*random(7..43#4),if(N>50,N+s3*random(7..43#4),N+random(2,4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n3": {"definition": "if(N>100,N+s4*random(6..42#4),if(N>50,N+s4*random(6..42#4),N+random(5,6)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n3", "description": ""}, "n4": {"definition": "if(N>100,N+s5*random(5..41#4),if(N>50,N+s5*random(5..41#4),N+random(7,8)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "ep": {"definition": "10^(-r)", "templateType": "anything", "group": "Ungrouped variables", "name": "ep", "description": ""}, "tval": {"definition": "(1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d)", "templateType": "anything", "group": "Ungrouped variables", "name": "tval", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>4/07/2012:</strong></p>\n        \t\t<p><strong><br /></strong>Changed inequality sign in prompt from $\\lt$ to $\\leq$ and as a consequence changed them in the Advice. Answer remains the same.</p>\n        \t\t<p><strong>21/07/2012:</strong></p>\n        \t\t<p>Added description.</p>\n        \t\t<p>Needs better tags to describe second part.</p>\n        \t\t<p>Also need to redefine the variables so that a and b &nbsp;and a and c are coprime - results in a better and less clumsy Advice solution. This is the \"changes needed\" tag. Issue raised as having defined a new function chcop using the gcd function, the editor did not register it in the variables list - although the question compiled and ran.</p>\n        \t\t<p>(Contd.) The variables a,b,c,d have been redefined. Also noticed that the MCQ had two correct answers on rare occasions. This has been corrected.</p>\n        \t\t<p>Got rid of the changes needed tag.</p>\n        \t\t<p><strong>27/7/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t", "description": "<p>Let $x_n=\\frac{an+b}{cn+d},\\;\\;n=1,\\;2\\ldots$. Find &nbsp;$\\lim_{x \\to\\infty} x_n=L$ and find least $N$ such that $|x_n-L| \\lt 10^{-r},\\;n \\geq N,\\;r \\in \\{2,\\;3,\\;4\\}$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Sequences and Limits 1.3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {"chcop": {"definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "type": "number", "language": "jme", "parameters": [["a", "number"], ["b", "number"]]}}, "ungrouped_variables": ["a", "c", "b", "r1", "something1", "s1", "m", "ep1", "b1", "d", "r", "something", "t", "n", "thisratio", "ep", "tval"], "tags": ["limit of a sequence", "limits", "sequence", "taking the limit"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>$\\begin{align}z&amp;=\\var{b}\\\\w&amp;=a\\end{align}$</p>\n<p>To find the least $N$ such that all terms from the the $N$th are within $10^{\\var{-r}}$ of the limit we proceed as follows:</p>\n<p>\\[\\begin{eqnarray} \\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\lt 10^{ -\\var{r}}&amp;\\iff &amp;\\left|\\simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}}\\right| \\lt 10^{ -\\var{r}}\\\\&amp;\\iff &amp;\\simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\\lt 10^{ -\\var{r}} \\end{eqnarray} \\]</p>\n<p>(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n+{c*d}} \\gt 0,\\;\\;\\forall n=1,\\;2,\\;3\\ldots$)</p>\n<p>Rearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n+{c*d}})10^{\\var{r}}$ (this is positive and so the inequality does not reverse) we get:<br/>\\[\\simplify[std]{{c^2}n+{c*d}} \\gt \\var{10^r*abs(b*c-a*d)} \\Leftrightarrow n \\gt \\frac{1}{\\var{c^2}}\\left(\\simplify[std]{{10^r*abs(b*c-a*d)}-{c*d}}\\right)=\\var{tval}\\]</p>\n<p>Hence the least integer value is given by rounding up i.e. $N=\\var{N}$.</p>\n<h4>b)</h4>\n<p>Using the same method you should obtain $N_1=\\var{m}$.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n<p>Find the least integer $N$ such that</p>\n<p>\\[\\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\lt 10 ^ { -\\var{r}},\\;\\;\\textrm{for}\\;\\;n \\geq N\\]</p>\n<p>Least $N=\\;\\;$[[0]]</p>\n", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{N}", "minValue": "{N}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n\n\n<p>Find the least integer $N_1$ such that</p>\n\n\n\n<p>\\[\\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\lt 10 ^ { \\var{-r+r1}},\\;\\;\\textrm{for}\\;\\;n \\geq N_1\\]</p>\n\n\n\n<p>Least $N_1=\\;\\;$[[0]]</p>\n\n\n", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{m}", "minValue": "{m}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Let \\[x_n=\\simplify[std]{({a}n+{b})/({c}n+{d})},\\;\\;n=1,\\;2,\\;3\\ldots\\]</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "chcop(a,a)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "if(b1*c=a*d,b1+1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "r1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "something": {"definition": "if(r1=1,'dividing','multiplying')", "templateType": "anything", "group": "Ungrouped variables", "name": "something", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "tval": {"definition": "(1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d)", "templateType": "anything", "group": "Ungrouped variables", "name": "tval", "description": ""}, "m": {"definition": "ceil((abs(b*c-a*d)-d*c*ep1)/(ep1*c^2))", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "ep1": {"definition": "10^(-r+r1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ep1", "description": ""}, "r": {"definition": "random(2,3,4,5,6)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "b1": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "t": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "n": {"definition": "ceil((abs(b*c-a*d)-d*c*ep)/(ep*c^2))", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "something1": {"definition": "if(r1=-1,'divides','multiplies')", "templateType": "anything", "group": "Ungrouped variables", "name": "something1", "description": ""}, "thisratio": {"definition": "if(r1=1, 'one tenth of ', '$10$ times ')", "templateType": "anything", "group": "Ungrouped variables", "name": "thisratio", "description": ""}, "ep": {"definition": "10^(-r)", "templateType": "anything", "group": "Ungrouped variables", "name": "ep", "description": ""}, "d": {"definition": "chcop(c,c)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>4/07/2012:</strong></p>\n        \t\t<p>Checked calculations.</p>\n        \t\t<p>Small changes to Advice display.</p>\n        \t\t<p>Left inequalities as $\\lt$.</p>\n        \t\t<p><strong>21/07/2012:</strong></p>\n        \t\t<p>Added description.</p>\n        \t\t<p>Added function chcop to create coprime pairs - better display of solution.</p>\n        \t\t<p>Changed definition of variables a, b, c, d.</p>\n        \t\t<p>&nbsp;<strong>27/7/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Question appears to be working correctly.</p>\n        \t\t", "description": "\n        \t\t<p>$x_n=\\frac{an+b}{cn+d}$.&nbsp;Find the least integer $N$ such that $\\left|x_n -\\frac{a}{c}\\right| \\lt 10 ^{-r},\\;n\\geq N$, $2\\leq r \\leq 6$.</p>\n        \t\t<p>&nbsp;</p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}