Find the common ratio for the following geometric series.
\n$\\var{a}, \\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\ldots$
\nCommon ratio: [[0]]
", "sortAnswers": false, "variableReplacements": [], "scripts": {}}, {"useCustomName": false, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "steps": [{"useCustomName": false, "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "customMarkingAlgorithm": "", "customName": "", "prompt": "The formula for the $n^\\text{th}$ term of a geometric sequence is
\n\\[ a_n = ar^{(n-1)} \\]
\nwhere $a$ is the first term in the sequence and $r$ is the common ratio.
", "variableReplacements": [], "scripts": {}}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "gaps": [{"useCustomName": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "marks": 1, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "customMarkingAlgorithm": "", "answer": "{a}*{r}^(n-1)", "showPreview": true, "customName": "", "failureRate": 1, "checkingType": "absdiff", "valuegenerators": [{"name": "n", "value": ""}], "vsetRange": [0, 1], "variableReplacements": [], "scripts": {}}], "customMarkingAlgorithm": "", "stepsPenalty": 0, "customName": "", "prompt": "Write down the formula for the $n^\\text{th}$ term in the sequence
\n$a_n = $ [[0]]
", "sortAnswers": false, "variableReplacements": [], "scripts": {}}, {"useCustomName": false, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "gaps": [{"useCustomName": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "unitTests": [], "correctAnswerStyle": "plain", "maxValue": "a*r^(n-1)", "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "allowFractions": false, "customMarkingAlgorithm": "", "minValue": "a*r^(n-1)", "correctAnswerFraction": false, "mustBeReduced": false, "customName": "", "showFractionHint": true, "variableReplacements": [], "scripts": {}}], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "customMarkingAlgorithm": "", "customName": "", "prompt": "What is the $\\var{n}^\\text{th}$ term in this sequence?
\n$a_\\var{n} =$ [[0]]
", "sortAnswers": false, "variableReplacements": [], "scripts": {}}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find the common ratio of a given geometric sequence, write down the formula for the nth term and use it to calculate a given term in the sequence.
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", "group": "Ungrouped variables", "definition": "random(3..10 except r)"}, "nth_term": {"name": "nth_term", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a*r^n"}, "n": {"name": "n", "templateType": "anything", "description": "The index of a term to calculate.
\nThe range is picked so that the number is between 1,000 and 1,000,000.
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", "group": "Ungrouped variables", "definition": "random(3..8)"}}, "name": "Justin's copy of Find and use the formula for a geometric sequence", "ungrouped_variables": ["a", "r", "n", "nth_term"], "advice": "The terms in a geometric sequence are found by repeatedly multiplying the last term by a constant, called the common ratio.
\nTo find the common ratio, pick a term of the sequence and divide it by the previous term.
\nWe can calculate the common ratio using a table:
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{a}$ | \n$\\var{a*r}$ | \n$\\var{a*r^2}$ | \n$\\var{a*r^3}$ | \n
| $a_n \\div a_{n-1}$ | \n\n | $\\var{r}$ | \n$\\var{r}$ | \n$\\var{r}$ | \n
The common ratio is $\\var{r}$.
\nThe general formula for the $n^\\text{th}$ term of a geometric sequence is
\n\\[\\displaystyle {a_n=ar^{(n-1)}\\text{,}}\\]
\nwhere $a$ is the first term, and $r$ is the common ratio.
\nSo the formula for this sequence is
\n\\[ a_n = \\simplify[]{ {a}*{r}^n } \\text{.} \\]
\nWe know from part b) that the formula for the $n^\\text{th}$ term is $a_n = \\simplify[]{ {a}*{r}^n}$.
\nTherefore the $\\var{n}^\\text{th}$ term in the sequence is
\n\\begin{align}
a_\\var{n} &= \\var{a} \\times \\var{r}^{\\var{b}} \\\\
&= \\var{a*r^n}
\\end{align}