// Numbas version: exam_results_page_options {"name": "Identificar diferentes tipos de sucesiones... ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Identificar diferentes tipos de sucesiones... ", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Una sucesión es una lista de números que siguen un patrón. Diferentes tipos de sucesiones siguen patrones diferentes, por ejemplo, algunos aumentan en una cantidad constante de término a término (PA) y otros siguen una regla de multiplicar el término anterior por una constante para encontrar el siguiente término (PG).

", "advice": "

a)

\n

La sucesión lineal es una sucesión con una diferencia común constante, mientras que una sucesión cuadrática tiene una diferencia que aumenta o disminuye en un valor constante.

\n

Podemos calcular las diferencias entre términos construyendo tablas. Estas tablas se pueden usar para decidir si la sucesión es lineal o cuadrática.

\n

i)

\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$$\\var{b[2]+n[2]+c[2]}$$\\var{b[2]*2^2+n[2]*2+c[2]}$$\\var{b[2]*3^2+n[2]*3+c[2]}$$\\var{b[2]*4^2+n[2]*4+c[2]}$
Diferencia entre términos$\\var{b[2]*2^2+n[2]*2+c[2]-b[2]*1^2-n[2]*1-c[2]}$$\\var{b[2]*3^2+n[2]*3+c[2]-b[2]*2^2-n[2]*2-c[2]}$$\\var{b[2]*4^2+n[2]*4+c[2]-b[2]*3^2-n[2]*3-c[2]}$
\n

Si observamos las diferencias con este patrón, vemos que aumentan en $\\var{2*b[2]}$ unidades. A medida que las diferencias originales en la secuencia aumentan en una constante, la secuencia es cuadrática.

\n

ii)

\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$$\\var{m[3]*5}$$\\var{m[3]*6}$$\\var{m[3]*7}$$\\var{m[3]*8}$
Diferencia entre términos$\\var{m[3]}$$\\var{m[3]}$$\\var{m[3]}$
\n


Esta sucesión se incrementa en $\\var{m[3]}$ unidades, por lo tanto es lineal.

\n

iii)

\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$$3$
$a_n$$\\var{m[2]*10}$$\\var{m[2]*11}$$\\var{m[2]*12}$
Diferencia entre términos$\\var{m[2]}$$\\var{m[2]}$
\n

iv)

\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$$\\var{b[4]*1^2+n[4]*1+c[4]}$$\\var{b[4]*2^2+n[4]*2+c[4]}$$\\var{b[4]*3^2+n[4]*3+c[4]}$$\\var{b[4]*4^2+n[4]*4+c[4]}$
Diferencia entre términos$\\var{3*b[4]+n[4]}$$\\var{5*b[4]+n[4]}$$\\var{7*b[4]+n[4]}$
\n

Si observamos las diferencias con este patrón, vemos que aumentan en $\\var{2*b[4]}$ unidades . A medida que las diferencias originales en la secuencia aumentan en una constante, la sucesión es cuadrática.

\n

b)

\n

Las sucesiones aritméticas se pueden identificar por el hecho de que tienen una diferencia común, mientras que las secuencias geométricas tienen una razón común.

\n

i)

\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$$\\var{c[0]}$$\\var{c[0]^2}$$ \\var{c[0]^3}$$ \\var{c[0]^4}$
Diferencia entre términos$\\var{c[0]^2-c[0]}$$\\var{c[0]^3-c[0]^2}$$\\var{c[0]^4-c[0]^3}$
\n

Esta secuencia tiene una razón común de $\\var{c[0]} $ por lo que la secuencia es geométrica.

\n

\n

ii)

\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$$\\var{c[2]*c[3]}$$\\var{c[2]*c[3]^2}$$ \\var{c[2]*c[3]^3}$$ \\var{c[2]*c[3]^4}$
Diferencia entre términos$\\var{c[2]*c[3]^2-c[2]*c[3]}$$\\var{c[2]*c[3]^3-c[2]*c[3]^2}$$\\var{c[2]*c[3]^4-c[2]*c[3]^3}$
\n

Esta secuencia tiene una razón común de $\\var{c[3]}$ por lo que la secuencia es geométrica.

\n

iii)

\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$$\\var{c[1]*5}$$\\var{c[1]*6}$$ \\var{c[1]*7}$$ \\var{c[1]*8}$
Diferencia entre términos$\\var{c[1]}$$\\var{c[1]}$$\\var{c[1]}$
\n

Cada término en esta secuencia tiene una diferencia común de $\\var{c [1]}$ por lo que la secuencia es aritmética.

\n

iv)

\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$$\\var{c[2]*8}$$\\var{c[2]*9}$$ \\var{c[2]*10}$$ \\var{c[2]*11}$
Diferencia entre términos$\\var{c[2]}$$\\var{c[2]}$$\\var{c[2]}$
\n

Cada término en esta secuencia tiene una diferencia común de $\\var{c[2]} $ por lo que la secuencia es aritmética.

\n

\n

c)

\n

Podemos usar una tabla para identificar la secuencia de triángulos por su característica: la diferencia entre los términos que aumentan en $ 1 $ con cada término.

\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\n\n\n\n\n\n\n\n
$n$$1$$2$$3$$4$$5$$6$$7$
$a_n$$1$$3$$6$$10$$a_5$$a_6$$a_7$
Diferencia entre términos$2$$3$$4$$5$$6$$7$
\n

Luego podemos usar este patrón para continuar la secuencia e identificar los siguientes tres términos ($a_5$, $a_6$ y $a_7$).

\n

\\[a_5=a_4+5=15\\]

\n

\\[a_6=a_5+6=21\\]

\n

\\[a_7=a_6+7=28\\]

\n

o

\n

Podemos usar la fórmula para el término $n^{th}$ de la secuencia del triángulo

\n

\\[\\frac{n(n+1)}{2}\\text{.}\\]

\n

Para encontrar los términos $5th, 6th\\; \\text{y}\\; 7th$.

\n

\\[\\begin{align}
\\frac{5(5+1)}{2}&=15\\\\
\\frac{6(6+1)}{2}&=21\\\\
\\frac{7(7+1)}{2}&=28
\\end{align}\\]

\n

d)

\n

Para encontrar la respuesta a esta pregunta, debemos sustituir nuestro valor por n=($\\var{ci[0]}$) en la fórmula para la secuencia que ya se nos ha dado:

\n

\\[a_n=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\text{.}\\]

\n

Por lo tanto:

\n

\\[
\\begin{align}
a_{\\var{ci[0]}}&=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\\\
&=\\frac{\\var{a1}\\times\\var{ci[0]}(\\var{ci[0]}+\\var{b1})}{\\var{c2}}\\\\
&=\\simplify{{{a1}*{ci[0]}*({ci[0]}+{b1})}/{c2}}
\\end{align}
\\]

\n

 

\n

e)

\n

i)

\n

Podemos analizar la secuencia utilizando una tabla para visualizar cada término con la diferencia entre sí y el término anterior.

\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\n\n
$n$$1$$2$$3$$4$$5$
$a_n$$1$$4$$9$$16$$25$
Diferencia entre términos$3$$5$$7$$9$
\n

Observe que la diferencia entre los términos aumenta con cada término, y al observar cuidadosamente la secuencia, podemos darnos cuenta de que esta es una secuencia cuadrada que significa que cada término se ha cuadrado para obtener su valor.

\n

\\[\\begin{align}
1^2&=1\\\\
2^2&=4\\\\
3^2&=9\\\\
4^2&=16\\\\
5^2&=25
\\end{align}\\]

\n

Por lo tanto para obtener los tres valores siguientes tenemos que cuadrar los valores.

\n

\\[\\begin{align}
6^2&=36\\\\
7^2&=49\\\\
8^2&=64\\text{.}
\\end{align}\\]

\n

ii)

\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$$1$$8$$27$$64$
Diferencia entre términos$7$$21$$37$
\n

Observe que la diferencia entre los términos aumenta con cada término, y al observar cuidadosamente la secuencia, podemos darnos cuenta de que se trata de una secuencia cúbica que significa que cada término se ha dividido en cubos para obtener su valor.

\n

\\[\\begin{align}
1^3&=1\\\\
2^3&=8\\\\
3^3&=27\\\\
4^3&=64
\\end{align}\\]

\n

Por lo tanto, para obtener los tres valores siguientes tenemos que calcular los valores

\n

\\[\\begin{align}
5^3&=125\\\\
6^3&=216\\\\
7^3&=343\\text{.}
\\end{align}\\]

\n

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Para cada una de estas secuencias indicar si son lineales o cuadráticas.

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ii) $\\var{m[3]*5}, \\var{m[3]*6}, \\var{m[3]*7}, \\var{m[3]*8}...$

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iii) $\\var{m[2]*10}, \\var{m[2]*11}, \\var{m[2]*12}...$

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iv) $\\var{b[4]*1^2+n[4]*1+c[4]}, \\var{b[4]*2^2+n[4]*2+c[4]}, \\var{b[4]*3^2+n[4]*3+c[4]}, \\var{b[4]*4^2+n[4]*4+c[4]}...$

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Cuadrática

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Lineal

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¿Cuáles de estas secuencias son aritméticas y cuáles son geométricas?

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i) $\\var{c[0]}, \\var{c[0]^2}, \\var{c[0]^3}, \\var{c[0]^4}\\ldots$

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ii) $\\var{c[2]*c[3]}, \\var{c[2]*c[3]^2}, \\var{c[2]*c[3]^3}, \\var{c[2]*c[3]^4}\\ldots$

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iii) $\\var{c[1]*5}, \\var{c[1]*6}, \\var{c[1]*7}, \\var{c[1]*8}, \\ldots$

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iv) $\\var{c[2]*8}, \\var{c[2]*9}, \\var{c[2]*10}, \\var{c[2]*11} \\ldots$

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Aritmética

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Geométrica

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Escribe los  tres  siguientes términos en la secuencia de los números triángulares.

\n

$1, 3, 6, 10,$ [[0]], [[1]], [[2]]

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El término $n^{th}$  de la secuencia de los números triangulares es 

\n

\\[\\frac{n(n+1)}{2}\\text{.}\\]

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Encontrar  el término $\\var{ci[0]}^{th}$ de la secuencia mediante la fórmula: 

\n

\\[a_n=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\]

\n

$a_{\\var{ci[0]}}=$ [[0]]

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Escribe los  tres términos siguientes en las secuencias.

\n

i)

\n

$1, 4, 9, 16, 25,$ [[0]], [[1]], [[2]]

\n

ii)

\n

$1, 8, 27, 64,$ [[3]], [[4]], [[5]]

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