// Numbas version: exam_results_page_options
{"name": "Practical Arithmetic Sequences 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Practical Arithmetic Sequences 2", "tags": [], "metadata": {"description": "Falling object used to investigate arithmetic sequence, extension sections relate to geometric sequences and percentage growth.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Prachi is on vacation in the United States. She is visiting the Grand Canyon.&nbsp;When she reaches the top, she drops a coin down a cliff. The coin falls down a distance of&nbsp;{u_1} metres during the first second,&nbsp;{u_2} metres during the next second,&nbsp;{u_3} metres during the third second and continues in this way. The distances that the coin falls during each second forms an arithmetic sequence.</p>", "advice": "<p>Parts (a) - (d) involve an arithmetic sequence.</p>\n<p>In part (a) the common difference is the difference between any two consecutive terms. Either $u_2 - u_1$ or&nbsp;$u_3 - u_2$. These both evaluate to give $d = 10 $ metres.</p>\n<p>In part (b) you can find the fourth term by adding d to the thr&igrave;ird term $u_4 = u_3 + d = \\var{u_3} + 10 = \\var{u_4}$.</p>\n<p>In part (c) you should apply the formula $u_n = u_1 + (n-1)d$.</p>\n<p>In this case you are finding the {nth} term so&nbsp;$u_\\var{n} = u_1 + (\\var{n}-1)d$</p>\n<p>$u_\\var{n} = \\var{u_1} + (\\var{n}-1) \\times \\var{d} = \\var{u_n}$</p>\n<p>In part (d) you are now summing terms so should use the formula for an arithmetic sequence $S_n = \\frac{n}{2}(2u_1 + (n-1)d)$ or alternatively since you have just found&nbsp;$u_\\var{n}$ you could use $S_n = \\frac{n}{2}(u_1 + u_n)$</p>\n<p>In this case $S_\\var{n} = \\frac{\\var{n}}{2}( \\var{u_1} + \\var{u_n}) = \\var{1000*S_n}$ metres which corresponds to $\\var{S_n}$ kilometres.</p>\n<p>In parts (e) and (f) you are given information about a geometric sequence with first term $\\var{visitors}$ and common ratio given by $1 + \\frac{\\var{p}}{100} = \\var{1 + p/100}$.</p>\n<p>In (e) the number of visitors in 2016 is $\\var{visitors} \\times&nbsp;\\var{1 + p/100} = \\var{visits_2016} $. You may have rounded this answer to 3 significant figures.&nbsp;</p>\n<p>In (f) you are now summing the first ten terms and should use the formula $S_n = \\frac&nbsp; {u_1(r^n -1 )}{r-1}$</p>\n<p>$S_{10} = \\frac{\\var{visitors} \\times (\\var{1 + p/100}^{10}-1)}{\\var{1 + p/100}-1}$</p>\n<p>This value should be rounded to 3 significant figures giving $S_{10}=\\var{visits_10y_rounded}$.</p>", "rulesets": {}, "extensions": [], "variables": {"u_1": {"name": "u_1", "group": "Ungrouped variables", "definition": "5", "description": "<p>first term</p>", "templateType": "number"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "10", "description": "<p>common difference</p>", "templateType": "number"}, "u_2": {"name": "u_2", "group": "Ungrouped variables", "definition": "u_1 + d", "description": "<p>second term</p>", "templateType": "anything"}, "u_3": {"name": "u_3", "group": "Ungrouped variables", "definition": "u_1 + 2*d", "description": "<p>third term</p>", "templateType": "anything"}, "u_4": {"name": "u_4", "group": "Ungrouped variables", "definition": "u_1 + 3d", "description": "<p>fourth term</p>", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(10 .. 20#1)", "description": "<p>number of terms to sum</p>", "templateType": "randrange"}, "u_n": {"name": "u_n", "group": "Ungrouped variables", "definition": "u_1 + (n-1)*d", "description": "<p>nth term</p>", "templateType": "anything"}, "S_n": {"name": "S_n", "group": "Ungrouped variables", "definition": "(n/2000)*(2*u_1 + (n-1)*d)", "description": "<p>sum of n terms</p>", "templateType": "anything"}, "number_eng": {"name": "number_eng", "group": "Ungrouped variables", "definition": "[ \"tenth\", \"elventh\", \"twelfth\", \"thirteenth\", \"fourteenth\", \"fifteenth\", \"sixteenth\", \"seventeenth\", \"eighteenth\", \"nineteenth\", \"twentieth\" ]", "description": "", "templateType": "list of strings"}, "nth": {"name": "nth", "group": "Ungrouped variables", "definition": "number_eng[n-10]", "description": "", "templateType": "anything"}, "visitors": {"name": "visitors", "group": "Ungrouped variables", "definition": "random(14000 .. 19000#1000)", "description": "<p>Number of visitors in 2015</p>", "templateType": "randrange"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(8 .. 12#1)", "description": "<p>p% increase in visitors expected.</p>", "templateType": "randrange"}, "visits_2016": {"name": "visits_2016", "group": "Ungrouped variables", "definition": "visitors*(1+p/100)", "description": "<p>visitors in 2016</p>", "templateType": "anything"}, "visits_10y": {"name": "visits_10y", "group": "Ungrouped variables", "definition": "visitors*(1 - (1 + p/100)^10)/(1- (1 + p/100))", "description": "<p>visitors in first ten years.</p>", "templateType": "anything"}, "min_visits_2016": {"name": "min_visits_2016", "group": "Ungrouped variables", "definition": "min(siground(visits_2016,3),visits_2016)", "description": "<p>minimum accepted answer to part (e) since result may be rounded or given exactly.</p>", "templateType": "anything"}, "max_visits_2016": {"name": "max_visits_2016", "group": "Ungrouped variables", "definition": "max(siground(visits_2016,3),visits_2016)", "description": "<p>maximum&nbsp;accepted answer to part (e) since result may be rounded or given exactly.</p>", "templateType": "anything"}, "visits_10y_rounded": {"name": "visits_10y_rounded", "group": "Ungrouped variables", "definition": "siground(visits_10y,3)", "description": "<p>10 year visits rounded to 3 significant figures.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["u_1", "d", "u_2", "u_3", "u_4", "n", "u_n", "S_n", "number_eng", "nth", "visitors", "p", "visits_2016", "visits_10y", "min_visits_2016", "max_visits_2016", "visits_10y_rounded"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the common difference (in metres) of this arithmetic sequence.</p>", "minValue": "d", "maxValue": "d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What distance (In metres) does the coin fall in the fourth second.</p>", "minValue": "u_4", "maxValue": "u_4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What distance (in metres) does the coin fall in the {nth} second.</p>", "minValue": "u_n", "maxValue": "u_n", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the distance in kilometres that the coin falls in the first {n} seconds.</p>", "minValue": "S_n", "maxValue": "S_n", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "75", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Prachi visits a tourist centre nearby. It opened at the start of 2015 and in the first year there were&nbsp;{visitors} visitors. The number of people who visit the tourist centre is expected to increase by {p}% each year.</p>"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the number of visitors expected to visit the tourist centre in 2016.</p>", "minValue": "min_visits_2016", "maxValue": "max_visits_2016", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the number of people expected to visit the tourist centre in the first ten years since it opened.</p>", "minValue": "visits_10y", "maxValue": "visits_10y", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "75", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Johnathan Gregg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4974/"}]}]}], "contributors": [{"name": "Johnathan Gregg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4974/"}]}