// Numbas version: exam_results_page_options {"name": "Find and use the formula for a geometric sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find and use the formula for a geometric sequence", "tags": ["common ratio", "geometric sequences", "nth term", "sequences", "taxonomy"], "metadata": {"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.

", "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.

\n

a)

\n

To find the common ratio, pick a term of the sequence and divide it by the previous term.

\n

We can calculate the common ratio using a table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$$1$$2$$3$$4$
$a_n$$\\var{a}$$\\var{a*r}$$\\var{a*r^2}$$\\var{a*r^3}$
$a_n \\div a_{n-1}$$\\var{r}$$\\var{r}$$\\var{r}$
\n

The common ratio is $\\var{r}$.

\n

b)

\n

The general formula for the $n^\\text{th}$ term of a geometric sequence is

\n

\\[\\displaystyle {a_n=ar^{(n-1)}\\text{,}}\\]

\n

where $a$ is the first term, and $r$ is the common ratio.

\n

So the formula for this sequence is

\n

\\[ a_n = \\simplify[]{ {a}*{r}^(n-1) } \\text{.} \\]

\n

c)

\n

We know from part b) that the formula for the $n^\\text{th}$ term is $a_n = \\simplify[]{ {a}*{r}^{(n-1)}}$. 

\n

Therefore 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}

", "rulesets": {}, "extensions": [], "variables": {"nth_term": {"name": "nth_term", "group": "Ungrouped variables", "definition": "a*r^n", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3..10 except r)", "description": "

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.

\n

The range is picked so that the number is between 1,000 and 1,000,000.

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "r", "n", "nth_term"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "

Find the common ratio for the following geometric series.

\n

$\\var{a}, \\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\ldots$

\n

Common 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}.

\n

Write 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)} \\]

\n

where $a$ is the first term in the sequence and $r$ is the common ratio.

"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}*{r}^(n-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n", "value": ""}]}], "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": "

What is the $\\var{n}^\\text{th}$ term in this sequence?

\n

$a_\\var{n} =$ [[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": "a*r^(n-1)", "maxValue": "a*r^(n-1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}]}