Chapter 12.1, Problem 16CFU

To determine

To calculate: Thenumber of seats in the section of theater.

Answer to Problem 16CFU

The number of seats in the section of theaterare $342$ .

Explanation of Solution

Given information:

The section of the theater has 19 rows and number of seats in last row are 27. As moved toward the back of the section the number of seats in each row is increased by 1.

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the section of the theater has 19 rows and number of seats in last row are 27. As moved toward the back of the section the number of seats in each row is increased by 1.

The above situation forms an arithmetic sequence as a constant increase in number of seats in each row.

Denote the number of seats in last row as last term of arithmetic sequence $a_{19}=27$ .

Here, $d=1\text{ and }n=19$

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

  $\begin{array}{c}27=a_1+\left(19-1\right)1\\ 27=a_1+18\\ a_1=9\end{array}$

Number of seats in first row are 9.

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ or $S_n=\frac{n}{2}\left[a_1+a_n\right]$

Here $a_1=9\text{ and }{a}_n=27$ .

Again apply the sum of n terms of the arithmetic sequence $S_n=\frac{n}{2}\left[a_1+a_n\right]$ ,

  $\begin{array}{c}{S}_{19}=\frac{19}{2}\left[9+27\right]\\ =\frac{19}{2}\left[36\right]\\ =342\end{array}$

Thus, the number of seats in the section of theater are $342$ .