Chapter 12.2, Problem 65AYU

Seats in an Amphitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

To determine

To find: An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

Answer to Problem 65AYU

1,647

Explanation of Solution

Given:

Number of seats in the first row $=35$.

Number of seats in the second row $=37$.

Number of seats in the third row $=39$.

Total number of rows $=27$.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_1+\left(n-1\right)d$   (Formula 1)

Sum of the first $n$ terms of an arithmetic sequence,

$S_n=\frac{n}{2}\left[a_1+a_n\right]$   (Formula 2)

Calculation:

From the given data, we get $a_1=\text{ 35},n=\text{ 27},d=2$.

To find the number of seats in the 27^th row:

$a_{27}=\text{ 35 }+\left(\text{26}\right)\left(2\right)=\text{ 87}$   (By formula 1)

Therefore, the number of seats in the 27^th row $=87$.

To find the total number of total number of seats in the amphitheater:

$S_n=\frac{27}{2}\left[\text{35 }+\text{ 87}\right]$   (By formula 2)

$=\text{ 1,647}$

Therefore, total number of seats in the amphitheater $=\text{ 1,647}$.