 # el diseño de un teatro consta de 27 sillas en la primera fila , 32 en la segunda, 37 en la tercera y asi sucesivamente. se sabe que el teatro tiene 10 filas. ¿ cuantas sillas tiene el teatro? Alyce Wilkinson 2021-02-08 Answered
el diseño de un teatro consta de 27 sillas en la primera fila , 32 en la segunda, 37 en la tercera y asi sucesivamente. se sabe que el teatro tiene 10 filas.
¿ cuantas sillas tiene el teatro?
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The number of chairs in each row is represented by the sequence
The total number of chairs is then represented by the series
Since $32-27=5$ and $37-32=5$, then the number of chairs in each row is represented by an arithmetic sequence with a first term of ${a}_{1}=27$ and a a common difference of $d=5$.
The total number of chairs will then be found using the partial arithmetic sum formula ${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$​. It is given that the theater has $10$ rows so $n=10$. To use this formula when $n=10$, you must then find ${a}_{10}$.
The nnth term of an arithmetic sequence is ${a}_{n}={a}_{1}+\left(n-1{\right)}^{d}$. Therefore:
${a}_{10}$
$=27+\left(10-1\right)\left(5\right)$
$=27+9\left(5\right)$
$=27+45$
$=72$
The total number of chairs is then:
${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$ Partial sum formula.
${S}_{10}=\frac{10\left(27+72\right)}{2}$ Substitute $n=10,{a}_{1}=27$ and ${a}_{1}0=72$
$=5\left(99\right)$
$495$
The theater then has $495$ chairs.