// Numbas version: exam_results_page_options {"name": "Hallar el t\u00e9rmino general de una secuencia geom\u00e9trica.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "rulesets": {}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

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.

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Encontrar la razón común para la siguiente serie geométrica.

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

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Razón común: [[0]]

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La fórmula para el término $n^\\text{th}$ de la secuencia geométrica es

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$a_n = ar^{(n-1)}$

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donde $a$ es el primer término en la secuencia $r$ es la razón común.

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Encontrar la fórmula para el término $n^\\text{th}$ de la secuencia

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$a_n =$ [[0]]

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¿Cuál es el término $\\var{n}^\\text{th}$ en esta secuencia?

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$a_\\var{n} =$ [[0]]

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The index of a term to calculate.

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

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The first term

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The common ratio

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The terms in a geometric sequence are found by repeatedly multiplying the last term by a constant, called the common ratio.

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#### a)

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

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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
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{a}$ $\\var{a*r}$ $\\var{a*r^2}$ $\\var{a*r^3}$ $a_n \\div a_{n-1}$ $\\var{r}$ $\\var{r}$ $\\var{r}$
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The common ratio is $\\var{r}$.

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#### b)

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

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

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

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So the formula for this sequence is

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\$a_n = \\simplify[]{ {a}*{r}^n } \\text{.} \$

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#### c)

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

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Therefore the $\\var{n}^\\text{th}$ term in the sequence is

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\\begin{align}
a_\\var{n} &= \\var{a} \\times \\var{r}^{\\var{b}} \\\\
&= \\var{a*r^n}
\\end{align}

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