// Numbas version: exam_results_page_options
{"name": "Sequences and Limits 1.3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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"], "name": "Sequences and Limits 1.3", "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": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}