Question

On the main floor of Kodak Hall at the Regal Theater, the number of seats per row increases at a constant rate. Steven counts 31 seats in row 3 and 37 seats in row 6. How many seats are there in row 20?

67

69

71

65
Notes

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Answer 1

To find the number of seats in row 20, we need to determine the pattern in the number of seats based on the information provided.

We know:
- Row 3 has 31 seats
- Row 6 has 37 seats

First, let's find the increase in the number of seats from row 3 to row 6.

The difference in row numbers is:
6 - 3 = 3

The difference in the number of seats is:
37 - 31 = 6

This indicates that for every increase of 1 row, the number of seats increases by:
\[
\frac{6 \text{ seats}}{3 \text{ rows}} = 2 \text{ seats per row}
\]

Now, we can express the number of seats in terms of rows:
Let \(n\) be the row number. Since we know row 3 has 31 seats, we can express the number of seats \(S(n)\) in row \(n\) as:
\[
S(n) = S(3) + 2(n - 3)
\]
Where \(S(3) = 31\).

Now let's find \(S(20)\):
\[
S(20) = S(3) + 2(20 - 3)
\]
\[
S(20) = 31 + 2(17)
\]
\[
S(20) = 31 + 34
\]
\[
S(20) = 65
\]

Thus, the number of seats in row 20 is **65**.