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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\nThe range is picked so that the number is between 1,000 and 1,000,000.
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", "definition": "random(3..8)"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "The first term
", "definition": "random(3..10 except r)"}}, "tags": [], "parts": [{"variableReplacements": [], "gaps": [{"correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "minValue": "r", "type": "numberentry", "maxValue": "r", "useCustomName": false, "mustBeReduced": false, "showFractionHint": true, "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "allowFractions": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 1}], "type": "gapfill", "useCustomName": false, "prompt": "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]]
", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}, {"variableReplacements": [], "stepsPenalty": 0, "type": "gapfill", "useCustomName": false, "steps": [{"variableReplacements": [], "type": "information", "useCustomName": false, "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.
", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "prompt": "Write down the formula for the $n^\\text{th}$ term in the sequence
\n$a_n = $ [[0]]
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\n$a_\\var{n} =$ [[0]]
", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "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}