// Numbas version: exam_results_page_options {"name": "Encontrar la f\u00f3rmula para el t\u00e9rmino $n^{\\text {th}}$ de una secuencia lineal", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"type": "gapfill", "showFeedbackIcon": true, "scripts": {}, "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "customMarkingAlgorithm": "", "steps": [{"showCorrectAnswer": true, "prompt": "

La fórmula para el término $n^{\\text{th}}$ de una secuencia aritmética es: \$a_n = a_1 + (n-1)d, \$

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donde

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• $a_n$ es el término $n ^\\text {th}$;
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• $a_1$ es el primer término en la secuencia;
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• $d$ es la diferencia común entre términos consecutivos.
• \n
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Para esta secuencia aritmética, ¿qué es $a_1$?

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¿Qué es $d$?

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$\\var{m[1]*2}, \\var{m[1]*3}, \\var{m[1]*4}, \\ldots$

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Término de lugar  $n^\\text{th}$ = [[0]]

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La fórmula para el término $n^{\\text {th}}$ de una secuencia aritmética es $a_n = a_1 + (n-1) d$
dónde
$a_n$ es el término $n ^ \\ text {th}$;
$a_1$ es el primer término en la secuencia;
$d$ es la diferencia común entre términos consecutivos.

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Para esta secuencia aritmética, ¿qué es $a_1$?

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¿Qué es $d$?

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$\\var{m[2]*8+2}, \\var{m[2]*7+2}, \\var{m[2]*6+2}...$

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Término $n^\\text{th}$ = [[0]]

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Dados los primeros tres términos de una secuencia, da una fórmula para el término $n^\\text {th}$.

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En la primera secuencia, $d$ es positivo. En la segunda secuencia, $d$ es negativo.

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Una secuencia lineal es una serie de números que aumenta o disminuye en una cantidad constante en cada paso.

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Encuentre fórmulas para el término $n^ {\\text{th}}$ para cada una de las siguientes secuencias lineales, donde se dan los valores para $n = 1 \\text {,} 2 \\text {,}3$:

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Ambas secuencias son lineales o aritméticas. Para encontrar fórmulas para estas secuencias necesitamos identificar sus primeros términos y diferencias comunes..

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#### a)

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La fórmula para el término $n^\\text {th}$ de una secuencia aritmética es: $a_n = a_1 + (n-1) d \\text {.}$

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$a_1$ es el primer término y $d$ la diferencia común entre términos consecutivos, que debemos identificar.

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Podemos encontrarlos dibujando una tabla de $a_n$ contra $n$, y las diferencias entre términos consecutivos.

\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 $a_n$ $\\pmb{\\var{m[1]*2}}$ $\\var{m[1]*3}$ $\\var{m[1]*4}$ Primeras diferencias $\\pmb{\\var{m[1]}}$ $\\pmb{\\var{m[1]}}$
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El primer término y la diferencia común se han resaltado en negrita; Podemos usar estos para escribir la fórmula de la secuencia.

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\\begin{align}
a_n &= a_1+(n-1)d \\\\
&= \\var{m[1]*2}+(n-1)\\times\\var{m[1]} \\\\
&= \\var{m[1]}n + \\var{m[1]}\\text{.}
\\end{align}

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#### b)

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De manera similar a la parte a), podemos identificar $a_1$ y $d$ para esta secuencia dibujando una tabla de $a_n$ contra $n$.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $a_n$ $\\pmb{\\var{m[2]*8+2}}$ $\\simplify{{m[2]}*7+2}$ $\\simplify{{m[2]}*6+2}$ Primeras diferencias $\\pmb{\\var{-m[2]}}$ $\\pmb{\\var{-m[2]}}$
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El primer término y la diferencia común se han resaltado en negrita; Podemos usar estos para formar la fórmula para la secuencia.

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\\begin{align}
a_n &=a_1+(n-1)d \\\\
&=\\var{m[2]*8+2}+(n-1)\\times\\var{-m[2]} \\\\
&=-\\var{m[2]}n + \\var{m[2]*9+2}\\text{.}
\\end{align}

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