// Numbas version: exam_results_page_options {"name": "Marte's copy of 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": [{"tags": ["common ratio", "geometric sequences", "nth term", "sequences", "taxonomy"], "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.
"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "name": "Marte's copy of Find and use the formula for a geometric sequence", "statement": "", "ungrouped_variables": ["a", "r", "n", "nth_term"], "functions": {}, "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}
The common ratio
"}, "a": {"group": "Ungrouped variables", "definition": "random(3..10 except r)", "templateType": "anything", "name": "a", "description": "The first term
"}, "n": {"group": "Ungrouped variables", "definition": "random(ceil(log(1000,r)-log(a,r))..floor(log(1000000,r)-log(a,r)))", "templateType": "anything", "name": "n", "description": "The index of a term to calculate.
\nThe range is picked so that the number is between 1,000 and 1,000,000.
"}, "nth_term": {"group": "Ungrouped variables", "definition": "a*r^n", "templateType": "anything", "name": "nth_term", "description": ""}}, "extensions": [], "preamble": {"js": "", "css": ""}, "rulesets": {}, "parts": [{"gaps": [{"scripts": {}, "minValue": "r", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "type": "numberentry", "mustBeReduced": false, "correctAnswerFraction": false, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "maxValue": "r", "showCorrectAnswer": true, "showFeedbackIcon": true}], "scripts": {}, "marks": 0, "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]]
", "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true}, {"gaps": [{"vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "scripts": {}, "variableReplacements": [], "vsetrange": [0, 1], "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "marks": 1, "variableReplacementStrategy": "originalfirst", "type": "jme", "showCorrectAnswer": true, "showFeedbackIcon": true, "answer": "{a}*{r}^(n-1)"}], "scripts": {}, "prompt": "Write down the formula for the $n^\\text{th}$ term in the sequence
\n$a_n = $ [[0]]
", "marks": 0, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "steps": [{"scripts": {}, "marks": 0, "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]]
", "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true}], "variable_groups": [], "type": "question", "contributors": [{"name": "Marte Br\u00e5talien", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2349/"}]}]}], "contributors": [{"name": "Marte Br\u00e5talien", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2349/"}]}