// Numbas version: exam_results_page_options {"name": "Calcular la suma parcial de una sucesi\u00f3n aritm\u00e9tica", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"partial_sum": {"group": "Ungrouped variables", "definition": "num_terms*(first_term+last_term)/2", "name": "partial_sum", "templateType": "anything", "description": "

Sum of the terms in the sequence.

"}, "first_term": {"group": "Ungrouped variables", "definition": "m*first_index", "name": "first_term", "templateType": "anything", "description": "

The first term in the sequence.

"}, "num_terms": {"group": "Ungrouped variables", "definition": "random(11..30)", "name": "num_terms", "templateType": "anything", "description": "

The number of terms in the sequence.

"}, "m": {"group": "Ungrouped variables", "definition": "random(2..10)", "name": "m", "templateType": "anything", "description": "

Common difference between terms

"}, "last_term": {"group": "Ungrouped variables", "definition": "m*last_index", "name": "last_term", "templateType": "anything", "description": "

The last term in the sequence.

"}, "last_index": {"group": "Ungrouped variables", "definition": "first_index+num_terms-1", "name": "last_index", "templateType": "anything", "description": "

The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.

\n

This is the index of the last term shown.

"}, "first_index": {"group": "Ungrouped variables", "definition": "random(6..14)", "name": "first_index", "templateType": "anything", "description": "

The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.

\n

This is the index of the first term shown.

Se nos da la secuencia aritmética

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\$\\var{m*first_index}, \\var{m*(first_index+1)}, \\var{m*(first_index+1)}, \\ldots, \\var{last_term} \\text{.} \$

\n

#### a)

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Sea $n$ el número de términos en la secuencia. Entonces $a_n = \\var{last_term}$.

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Para encontrar $n$, debemos reorganizar la fórmula para el término $n^\\text{th}$

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\$a_n=a_1+(n-1)d\\text{.}\$

\n

El primer término es $a_1 = \\var{first_term}$ y la diferencia común es $d = \\var{first_term+m} - \\var{first_term} = \\var{m}$.

\n

Entonces tenemos

\n

\\begin{align}
d&=\\var{m} \\text{,} \\\\
a_1&= \\var{first_term} \\text{,} \\\\
a_n&=\\var{last_term} \\text{.}
\\end{align}

\n

Ahora sustituimos estos valores en la fórmula del término  $n^\\text{th}$ y reorganizamos para encontrar $n$.

\n

\\begin{align}
\\var{last_term}&=\\var{first_term}+\\var{m}(n-1) \\\\
\\var{last_term}-\\var{first_term}&=\\var{m}(n-1) \\\\
\\frac{\\var{last_term-first_term}}{\\var{m}}&=(n-1) \\\\
n&=\\var{(last_term-first_term)/m}+1 = \\var{num_terms}\\text{.}
\\end{align}

\n

#### b)

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La suma parcial de los primeros $n$ términos de una secuencia aritmética está dada por

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\$\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\$

\n

Sabemos de la parte a) que

\n

\\begin{align}
n&= \\var{num_terms} \\text{,} \\\\
a_1&= \\var{first_term} \\text{,} \\\\
a_n&= \\var{last_term} \\text{.}
\\end{align}

\n

Sustituimos estos valores en la fórmula, obteniendo

\n

\\begin{align}
\\sum\\limits_{i=1}^n{a_i}&=\\frac{n}{2}(a_1+a_n)\\\\
&= \\frac{\\var{num_terms}}{2}(\\var{first_term}+\\var{last_term})\\\\
&= \\simplify[]{ ({num_terms}*({first_term+last_term}))/2} \\\\
&= \\var{partial_sum} \\text{.}
\\end{align}

Dados los primeros y últimos términos de una secuencia aritmética finita, calcule el número de elementos y luego la suma de la secuencia.

\n

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La fórmula para el $n^\\text{th}$ término en una secuencia aritmética es

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\$a_n=a_1+(n-1)d\\text{,}\$

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donde

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

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Cual es el valor de $d$?

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¿Cuántos términos hay en esta secuencia?

\n

$n =$ [[0]]

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La suma de una secuencia aritmética $a_1, a_2, \\ldots, a_n$ es calculada usando la fórmula

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\$\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n) \$

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donde

\n
\n
• $a_1$ es primer término;
• \n
• $a_n$ es el $n^{th}$ término;
• \n
• $n$ es el número de términos.
• \n
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Encuentra la suma de la secuencia incluyendo el término $\\var{last_term}$.

\n

[[0]]

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$\\var{m*first_index}, \\var{m*(first_index+1)}, \\var{m*(first_index+2)}, \\ldots, \\var{last_term}.$