// 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": [{"name": "Finding the formula for the $n^{\\text{th}}$ term of linear sequences", "type": "question", "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:
", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"ni": {"group": "Ungrouped variables", "name": "ni", "description": "", "templateType": "anything", "definition": "repeat(random(19..40),10)"}, "c": {"group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything", "definition": "repeat(random(3..13 except[10]),8)"}, "n": {"group": "Ungrouped variables", "name": "n", "description": "", "templateType": "anything", "definition": "repeat(random(1..4),7)"}, "m": {"group": "Ungrouped variables", "name": "m", "description": "", "templateType": "anything", "definition": "repeat(random(2..10),5)"}, "b": {"group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything", "definition": "repeat(random(2..4), 5)"}, "ci": {"group": "Ungrouped variables", "name": "ci", "description": "", "templateType": "anything", "definition": "repeat(random(6..20),10)"}}, "extensions": [], "functions": {}, "tags": ["arithmetic formula", "common difference", "first term", "formula for the nth term", "linear sequences", "nth term", "sequences", "taxonomy"], "variable_groups": [], "parts": [{"variableReplacements": [], "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answer": "{m[1]}*n+{m[1]}", "marks": "3", "variableReplacements": [], "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "$\\var{m[1]*2}, \\var{m[1]*3}, \\var{m[1]*4}, \\ldots$
\n$n^\\text{th}$ term = [[0]]
", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "The formula for the $n^{\\text{th}}$ term of an arithmetic sequence is
\n\\[a_n = a_1 + (n-1)d, \\]
\nwhere
\nFor this arithmetic sequence, what is $a_1$?
", "minValue": "{m[1]*2}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[1]}", "showFeedbackIcon": true, "prompt": "What is $d$?
", "minValue": "{m[1]}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "marks": 0}, {"variableReplacements": [], "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answer": "-{m[2]}*n+{m[2]*9+2}", "marks": "3", "variableReplacements": [], "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "$\\var{m[2]*8+2}, \\var{m[2]*7+2}, \\var{m[2]*6+2}...$
\n$n^\\text{th}$ term = [[0]]
", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "The formula for the $n^{\\text{th}}$ term of an arithmetic sequence is
\n\\[a_n = a_1 + (n-1)d, \\]
\nwhere
\nFor this arithmetic sequence, what is $a_1$?
", "minValue": "{m[2]*8+2}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{-m[2]}", "showFeedbackIcon": true, "prompt": "What is $d$?
", "minValue": "{-m[2]}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "marks": 0}], "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Given the first three terms of a sequence, give a formula for the $n^\\text{th}$ term.
\nIn the first sequence, $d$ is positive. In the second sequence, $d$ is negative.
"}, "preamble": {"css": "", "js": ""}, "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.
\nThe formula for the $n^\\text{th}$ term of an arithmetic sequence is
\n\\[ a_n = a_1 + (n-1)d \\text{.} \\]
\n$a_1$ is the first term and $d$ the common difference between consecutive terms, which we need to identify.
\nWe can find these by drawing up a table of $a_n$ against $n$, and the differences between consecutive terms.
\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; we can use these to write the formula for the sequence.
\n\\begin{align}
a_n &= 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}
Similar to the part a), we can identify $a_1$ and $d$ for this sequence by drawing a table of $a_n$ against $n$.
\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; we can use these to form the formula for the sequence.
\n\\begin{align}
a_n &=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}