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?

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?

Question
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?

Answers (1)

2021-02-09
The number of chairs in each row is represented by the sequence 27, 32, 37, ....
The total number of chairs is then represented by the series 27 + 32 + 37 + ....
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 a1=27 and a a common difference of d=5.
The total number of chairs will then be found using the partial arithmetic sum formula Sn=n(a1+an)/2​. It is given that the theater has 10 rows so n=10. To use this formula when n=10, you must then find a10.
The nnth term of an arithmetic sequence is an=a1+(n−1)d. Therefore:
a10
=27+(10−1)(5)
=27+9(5)
=27+45
=72
The total number of chairs is then:
\(\displaystyle{S}{n}={\frac{{{n}{\left({a}{1}+{a}{n}\right)}}}{{{2}}}}\) Partial sum formula.
\(\displaystyle{S}{10}={\frac{{{10}{\left({27}+{72}\right)}}}{{{2}}}}\) Substitute n=10, a1=27 and a10=72
=5(99)
495
The theater then has 495 chairs.
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