Incorrect answer
", "name": "u2"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(c,c)", "description": "", "name": "d"}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u1=t1,v1,v2)", "description": "", "name": "w1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1*c=a*d,b1+1,b1)", "description": "", "name": "b"}, "t2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Decreasing\"", "description": "", "name": "t2"}, "t3": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Neither\"", "description": "", "name": "t3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..20)", "description": "", "name": "a"}, "mono": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c>b/d,1,2)", "description": "If a/c > b/d, the sequence is increasing. If a/c < b/d, the sequence is decreasing. a,b,c,d are chosen so that $\\dfrac{a}{c} \\neq \\dfrac{b}{d}$.
", "name": "mono"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "b1"}, "t1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Increasing\"", "description": "", "name": "t1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(isint(tval), tval +1,ceil(tval))", "description": "", "name": "n"}, "v2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"decreasing\"", "description": "", "name": "v2"}, "v1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"increasing\"", "description": "", "name": "v1"}, "u1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t1,t2)", "description": "Correct answer
", "name": "u1"}, "tval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt((1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d))", "description": "", "name": "tval"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(a,a)", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "s1", "b1", "d", "r", "n", "tval", "mono", "t1", "t2", "t3", "u1", "u2", "v1", "v2", "w1"], "name": "When does a sequence get within $d$ of its limit?", "functions": {"chcop": {"type": "number", "language": "jme", "definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "parameters": [["a", "number"], ["b", "number"]]}}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "Find the limit $\\ell$ of $\\{x_n\\}$. Input as a fraction or an integer.
\nLimit $\\ell=$ [[0]]
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a}/{c}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "marks": 2, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "Find the least integer $N$ such that
\n\\[\\left|{x_n -\\ell}\\right| < 10 ^ { -\\var{r}}, \\quad \\text{for } n \\geq N\\]
\nLeast $N=$ [[0]]
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "{n}", "maxValue": "{n}", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "variableReplacements": [], "marks": "8", "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"displayType": "radiogroup", "choices": ["{u1}
", "{u2}
", "{t3}
"], "showCorrectAnswer": true, "displayColumns": "3", "unitTests": [], "prompt": "Which one of the following describes $\\{x_n\\}$?
", "customMarkingAlgorithm": "", "distractors": ["", "", ""], "variableReplacements": [], "shuffleChoices": true, "showFeedbackIcon": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "matrix": ["2", 0, 0], "marks": 0}], "statement": "Let
\n\\[x_n=\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})}, \\quad n=1,2,3, \\ldots\\]
", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "Limits", "limits", "query", "sequences", "taking the limit", "tested1", "udf"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "$x_n=\\frac{an^2+b}{cn^2+d}$. Find the least integer $N$ such that $\\left|x_n -\\frac{a}{c}\\right| < 10 ^{-r},\\;n\\geq N$, $2\\leq r \\leq 6$. Determine whether the sequence is increasing, decreasing or neither.
"}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "To begin with, the limit $\\ell$ is obtained by dividing top and bottom by $n^2$:
\n\\[\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}} = \\simplify[std]{({a}+{b}/n^2) /({c}+{d}/n^2)} \\to \\simplify[std]{({a})/({c})}\\] as $n \\to \\infty$, so $\\displaystyle \\ell= \\simplify[std]{{a}/{c}}$.
\n\\begin{align}
\\left|\\simplify[std]{x_n -({a} / {c})}\\right| < 10 ^ { -\\var{r}} &\\iff \\left|\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}}\\right| < 10 ^ { -\\var{r}} \\\\
&\\iff \\simplify[std]{abs({b*c-a*d})/({c^2}n^2+{c*d})} <10 ^ { -\\var{r}}
\\end{align}
(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n^2+{c*d}} \\gt 0$, $\\forall n=1,2,3,\\ldots$)
\nRearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n^2+{c*d}})10^{\\var{r}}$ (this is positive and so the inequality does not reverse), we get:
\n\\[\\simplify[std]{{c^2}n^2+{c*d}} > \\var{10^r*abs(b*c-a*d)} \\iff n^2 > \\frac{1}{\\var{c^2}}\\left(\\simplify[std]{{10^r*abs(b*c-a*d)}-{c*d}}\\right)=\\var{tval^2} \\iff n> \\var{tval}\\]
\nHence the least integer value is given by $N=\\var{N}$.
\nGiven $x_n = \\dfrac{an^2+b}{cn^2+d}, c \\gt 0, d\\gt 0$ it can be shown that $x_n \\leq x_{n+1} \\iff \\dfrac{b}{d} \\leq \\dfrac{a}{c}$. Here $\\dfrac{b}{d}=\\dfrac{\\var{b}}{\\var{d}}$ and $\\dfrac{a}{c}=\\dfrac{\\var{a}}{\\var{c}}$. Therefore the sequence will be increasing if $\\dfrac{\\var{b}}{\\var{d}} \\leq \\dfrac{\\var{a}}{\\var{c}} $ and decreasing if $\\dfrac{\\var{b}}{\\var{d}} \\geq \\dfrac{\\var{a}}{\\var{c}} $. Hence the sequence is $\\var{w1}$.
", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}