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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.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "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-1) } \\text{.} \\]
\nWe know from part b) that the formula for the $n^\\text{th}$ term is $a_n = \\simplify[]{ {a}*{r}^{(n-1)}}$.
\nTherefore the $\\var{n}^\\text{th}$ term in the sequence is
\n\\begin{align}
a_\\var{n} &= \\var{a} \\times \\var{r}^{\\var{n-1}} \\\\
&= \\var{a*r^(n-1)}
\\end{align}
The first term
", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "The common ratio
", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(ceil(log(1000,r)-log(a,r))..floor(log(1000000,r)-log(a,r)))", "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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\n$\\var{a}, \\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\ldots$
\nCommon ratio: [[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "r", "maxValue": "r", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The first term in the sequence is {a}.
\nWrite down the formula for the $n^\\text{th}$ term in the sequence
\n$a_n = $ [[0]]
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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.
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\n$a_\\var{n} =$ [[0]]
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