Question

A theater has 15 seats in the front row. The number of seats in each row follows an arithmetic series with 72 seats in the last row. The theater has a total of 870 seats. How many rows does the theater have? By how many seats does each row increase? Please show all work.

Answer 1

Explanation:

Given:

Theater has 15 seats

Last Row has 72 seats.

Theater has a total of 870 seats.

Unknown: Number of rows

Number of seats in each row.

Equations: Since we know the total number of seats the Theater has we can use the sum of arithmetic series

`s = (a _(1) + a _(n) )/(2) n`

Where a1 is the first row of seats

s is total number of seats

an is last row of seats

n is number of rows.

`870 = (15 + 72)/(2) (n)`

`870 = (87)/(2) n`

`2(870) = 87n`

`20 = n`

So we have 20 rows.

To find how many seats does each row increase, we use this formula,

`a _(n) = a + (n - 1)d`

Let use the 20th row as an example,

`a _(20) = 15 + (19)d`

`72 = 15 + (19)d`

`57 = 19d`

`3 = d`

So the common difference is 3.

So the seats of each row increase by 3.