// Numbas version: exam_results_page_options {"name": " Finding the Formula for the $n^{\\text{th}}$ Term of Linear Sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "
Both of these sequences are linear, or arithmetic, sequences. To find formulas for these sequences we need to identify their first terms and common differences.
\nWe can use a table and the equation: \\[a_n=a_1+(n-1)d\\text{;}\\] with $a_1$ being the first term and $d$ the common difference; to find the formula for the sequence:
\n$n$ | \n1 | \n2 | \n3 | \n
$a_n$ | \n$\\pmb{\\var{m[1]*2}}$ | \n$\\var{m[1]*3}$ | \n$\\var{m[1]*4}$ | \n
First differences | \n\n | $\\pmb{\\var{m[1]}}$ | \n$\\pmb{\\var{m[1]}}$ | \n
The first term and common difference have been highlighted in bold and we can use these to form the formula for the sequence.
\n\\[ \\begin{align}n^{\\text{th}}\\text{term} &=a_1+(n-1)d\\\\ &=\\var{m[1]*2}+(n-1)\\times\\var{m[1]}\\\\&=\\var{m[1]}n + \\var{m[1]}\\text{.}\\end{align}\\]
\n\n
We can use a table and the equation: \\[a_n=a_1+(n-1)d\\text{;}\\] with $a_1$ being the first term and $d$ the common difference; to find the formula for the sequence:
\n$n$ | \n1 | \n2 | \n3 | \n
$a_n$ | \n$\\pmb{\\var{m[2]*8+2}}$ | \n$\\simplify{{m[2]}*7+2}$ | \n$\\simplify{{m[2]}*6+2}$ | \n
First differences | \n\n | $\\pmb{\\var{-m[2]}}$ | \n$\\pmb{\\var{-m[2]}}$ | \n
The first term and common difference have been highlighted in bold and we can use these to form the formula for the sequence.
\n\\[ \\begin{align}n^{\\text{th}}\\text{term} &=a_1+(n-1)d\\\\ &=\\var{m[2]*8+2}+(n-1)\\times\\var{-m[2]}\\\\&=-\\var{m[2]}n + \\var{m[2]*9+2}\\text{.}\\end{align}\\]
\n", "preamble": {"js": "", "css": ""}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "prompt": "
$\\var{m[1]*2}, \\var{m[1]*3}, \\var{m[1]*4}...$
\n$n^{th}$ term = [[0]]
\n", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "vsetrange": [0, 1], "type": "jme", "checkingtype": "absdiff", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "expectedvariablenames": [], "showpreview": true, "checkvariablenames": false, "scripts": {}, "answer": "{m[1]}*n+{m[1]}"}], "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "stepsPenalty": 0, "steps": [{"showCorrectAnswer": true, "scripts": {}, "prompt": "The arithmetic formula is
\n\\[a_n = a_1 + (n-1)d, \\]
\nwhere
$a_n$ - The n$^{th}$ term in an arithmetic sequence
$a_1$ - The $1^{st}$ term in an arithmetic sequence
$n$ - Term number
$d$ - The common difference.
For this arithmetic sequence, what is $a_1$?
", "marks": 1, "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "minValue": "{m[1]}", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "maxValue": "{m[1]}", "variableReplacements": [], "prompt": "What is $d$?
", "marks": 1, "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerStyle": "plain"}]}, {"showCorrectAnswer": true, "scripts": {}, "prompt": "$\\var{m[2]*8+2}, \\var{m[2]*7+2}, \\var{m[2]*6+2}...$
\n$n^\\text{th}$ term = [[0]]
", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "vsetrange": [0, 1], "type": "jme", "checkingtype": "absdiff", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "expectedvariablenames": [], "showpreview": true, "checkvariablenames": false, "scripts": {}, "answer": "-{m[2]}*n+{m[2]*9+2}"}], "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "stepsPenalty": 0, "steps": [{"showCorrectAnswer": true, "scripts": {}, "prompt": "The arithmetic formula is
\n\\[a_n = a_1 + (n-1)d, \\]
\nwhere
$a_n$ - The n$^{th}$ term in an arithmetic sequence
$a_1$ - The $1^{st}$ term in an arithmetic sequence
$n$ - Term number
$d$ - The common difference.
For this arithmetic sequence, what is $a_1$?
", "marks": 1, "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "minValue": "{-m[2]}", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "maxValue": "{-m[2]}", "variableReplacements": [], "prompt": "What is $d$?
", "marks": 1, "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerStyle": "plain"}]}], "variablesTest": {"maxRuns": 100, "condition": ""}, "name": " Finding the Formula for the $n^{\\text{th}}$ Term of Linear Sequences", "functions": {}, "metadata": {"description": "Question assessing the students understanding of linear sequences.
\nStudents are assessed on their ability to find the common difference and first term in a linear sequence and then find the nth term of the sequence using the arithmetic formula.
\n", "licence": "Creative Commons Attribution 4.0 International"}, "variables": {"b": {"definition": "repeat(random(2..4), 5)", "group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything"}, "n": {"definition": "repeat(random(1..4),7)", "group": "Ungrouped variables", "name": "n", "description": "", "templateType": "anything"}, "ci": {"definition": "repeat(random(6..20),10)", "group": "Ungrouped variables", "name": "ci", "description": "", "templateType": "anything"}, "m": {"definition": "repeat(random(2..10),5)", "group": "Ungrouped variables", "name": "m", "description": "", "templateType": "anything"}, "c": {"definition": "repeat(random(3..13 except[10]),8)", "group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything"}, "ni": {"definition": "repeat(random(19..40),10)", "group": "Ungrouped variables", "name": "ni", "description": "", "templateType": "anything"}}, "statement": "A linear sequence is a series of numbers that either increases or decreases by a constant amount at each step.
\nFind formulas for the $n^{\\text{th}}$ term for each of the following linear sequences, where the values for $n=1\\text{,}2\\text{,}3$ are given:
", "variable_groups": [], "extensions": [], "tags": ["arithmetic formula", "common difference", "first term", "formula for the nth term", "linear sequences", "nth term", "sequences"], "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "rulesets": {}, "type": "question", "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}]}]}], "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}]}