Encuentre la razón común de una secuencia geométrica dada, escriba la fórmula para el enésimo término y úsela para calcular un término dado en la secuencia.
"}, "parts": [{"sortAnswers": false, "prompt": "Encontrar la razón común para la siguiente serie geométrica.
\n$\\var{a}, \\var{a*r}, \\var{a*r^2}, \\var{a*r^3}, \\ldots$
\nRazón común: [[0]]
", "gaps": [{"minValue": "r", "variableReplacements": [], "correctAnswerStyle": "plain", "maxValue": "r", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "unitTests": [], "type": "numberentry", "correctAnswerFraction": false, "mustBeReduced": false, "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"]}], "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": []}, {"sortAnswers": false, "variableReplacements": [], "gaps": [{"failureRate": 1, "unitTests": [], "variableReplacements": [], "vsetRange": [0, 1], "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "marks": 1, "type": "jme", "answer": "{a}*{r}^(n-1)", "scripts": {}, "showCorrectAnswer": true, "checkVariableNames": false, "showPreview": true, "customMarkingAlgorithm": "", "expectedVariableNames": []}], "steps": [{"variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "information", "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "La fórmula para el término $n^\\text{th}$ de la secuencia geométrica es
\n$ a_n = ar^{(n-1)}$
\ndonde $a$ es el primer término en la secuencia $r$ es la razón común.
", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": []}], "marks": 0, "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "prompt": "Encontrar la fórmula para el término $n^\\text{th}$ de la secuencia
\n$a_n = $ [[0]]
", "customMarkingAlgorithm": "", "unitTests": []}, {"sortAnswers": false, "prompt": "¿Cuál es el término $\\var{n}^\\text{th}$ en esta secuencia?
\n$a_\\var{n} =$ [[0]]
", "gaps": [{"minValue": "a*r^(n-1)", "variableReplacements": [], "correctAnswerStyle": "plain", "maxValue": "a*r^(n-1)", "allowFractions": false, "marks": 1, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "unitTests": [], "type": "numberentry", "correctAnswerFraction": false, "mustBeReduced": false, "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"]}], "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": []}], "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"n": {"group": "Ungrouped variables", "description": "The index of a term to calculate.
\nThe range is picked so that the number is between 1,000 and 1,000,000.
", "templateType": "anything", "name": "n", "definition": "random(ceil(log(1000,r)-log(a,r))..floor(log(1000000,r)-log(a,r)))"}, "a": {"group": "Ungrouped variables", "description": "The first term
", "templateType": "anything", "name": "a", "definition": "random(3..10 except r)"}, "r": {"group": "Ungrouped variables", "description": "The common ratio
", "templateType": "anything", "name": "r", "definition": "random(3..8)"}, "nth_term": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "nth_term", "definition": "a*r^n"}}, "functions": {}, "name": "Hallar el t\u00e9rmino general de una secuencia geom\u00e9trica.", "variable_groups": [], "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}