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 $a_1$?
", "maxValue": "{m[1]*2}"}, {"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "{m[1]}", "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "unitTests": [], "type": "numberentry", "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "showFractionHint": true, "customName": "", "scripts": {}, "allowFractions": false, "customMarkingAlgorithm": "", "prompt": "What is $d$?
", "maxValue": "{m[1]}"}], "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "stepsPenalty": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [{"name": "n", "value": ""}], "scripts": {}, "marks": "3", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "{m[1]}*n+{m[1]}", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}], "unitTests": [], "type": "gapfill", "prompt": "$\\var{m[1]*2}, \\var{m[1]*3}, \\var{m[1]*4}, \\ldots$
\n$n^\\text{th}$ term = [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "unitTests": [], "type": "information", "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 $a_1$?
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", "maxValue": "{-m[2]}"}], "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "stepsPenalty": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [{"name": "n", "value": ""}], "scripts": {}, "marks": "3", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "-{m[2]}*n+{m[2]*9+2}", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}], "unitTests": [], "type": "gapfill", "prompt": "$\\var{m[2]*8+2}, \\var{m[2]*7+2}, \\var{m[2]*6+2}...$
\n$n^\\text{th}$ term = [[0]]
"}], "functions": {}, "name": "Finding the formula for the $n^{\\text{th}}$ term of linear sequences", "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "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.
"}, "extensions": [], "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:
", "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}