// Numbas version: exam_results_page_options {"name": "Write down and apply the formula for an arithmetic sequence.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "name": "Write down and apply the formula for an arithmetic sequence.", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "type": "jme", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "answer": "{a1}+(n-1){d}", "showFeedbackIcon": true, "answersimplification": "basic", "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "vsetrangepoints": 5, "expectedvariablenames": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "prompt": "
Write out an expression for $a_n$, the $n^{\\text{th}}$ term of the sequence, in terms of $n$.
\n$a_n =$ [[0]]
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Find the $\\var{small}^{\\text{th}}$ term
\n$a_{\\var{small}} = $ [[0]]
\n"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "a1+(large-1)*d", "maxValue": "a1+(large-1)*d", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "prompt": "
Find the $\\var{large}^{\\text{th}}$ term
\n$a_{\\var{large}} = $[[0]]
"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.
"}, "tags": ["arithmetic sequences", "nth term", "sequences", "taxonomy"], "variables": {"large": {"templateType": "anything", "description": "A large index to compute
", "definition": "random(10..50#5)*10", "name": "large", "group": "Ungrouped variables"}, "small": {"templateType": "anything", "description": "A small index to compute
", "definition": "random(6..10)", "name": "small", "group": "Ungrouped variables"}, "a1": {"templateType": "anything", "description": "The first term in the sequence
", "definition": "random(1..90)", "name": "a1", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "description": "", "definition": "random(3..13 except 10)", "name": "d", "group": "Ungrouped variables"}}, "rulesets": {}, "extensions": [], "functions": {}, "ungrouped_variables": ["a1", "d", "small", "large"], "statement": "In this question, consider the sequence
\n\\[ a = \\var{a1}, \\; \\var{a1+d}, \\; \\var{a1+d*2}, \\; \\var{a1+d*3}, \\; \\ldots \\]
\nA helpful person has drawn out a table of the terms so far.
\n$\\boldsymbol{n}$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n$\\ldots$ | \n
---|---|---|---|---|---|
$\\boldsymbol{a_n}$ | \n$\\var{a1}$ | \n$\\var{a1+d}$ | \n$\\var{a1+2d}$ | \n$\\var{a1+3d}$ | \n$\\ldots$ | \n
The formula for the $n^\\text{th}$ term, $a_n$, of an arithmetic sequence is
\n\\[ a_n=a_1+(n-1)d \\text{.} \\]
\n$a_1$ is the first term, and $d$ is the common difference between adjacent terms.
\nIn the given sequence, the common difference is $\\var{a1+d} - \\var{a1} = \\var{d}$, and the first term is $\\var{a1}$.
\nSo, the formula for this sequence is
\n\\[ a_n = \\var{a1} + (n-1) \\times \\var{d} \\text{.} \\]
\n\\[ a_\\var{small} = \\var{a1} + (\\var{small}-1) \\times \\var{d} = \\var{a1+(small-1)*d} \\text{.} \\]
\n\\[ a_\\var{large} = \\var{a1} + (\\var{large}-1) \\times \\var{d} = \\var{a1+(large-1)*d} \\text{.} \\]
\n