Question
A theater has 41 seats in the first row, 44 seats in the second row, 47 seats in the third row, and so on. How many seats are there in row 15?
Question content area bottom
Part 1
There are
enter your response here seats in row 15.
Question content area bottom
Part 1
There are
enter your response here seats in row 15.
Answers
GPT-4o mini
To find the number of seats in row 15, we can observe the pattern in the number of seats per row:
- Row 1: 41 seats
- Row 2: 44 seats
- Row 3: 47 seats
From this, we can see that each subsequent row increases by 3 seats. This indicates that the number of seats in each row forms an arithmetic sequence where:
- First term (Row 1): \( a_1 = 41 \)
- Common difference: \( d = 3 \)
The formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
Where:
- \( a_n \) is the number of seats in row \( n \)
- \( a_1 \) is the number of seats in the first row
- \( d \) is the common difference
- \( n \) is the row number
Plugging in the values for row 15:
\[
a_{15} = 41 + (15-1) \cdot 3
\]
Calculating it:
\[
a_{15} = 41 + 14 \cdot 3
\]
\[
a_{15} = 41 + 42
\]
\[
a_{15} = 83
\]
So, there are **83 seats** in row 15.
- Row 1: 41 seats
- Row 2: 44 seats
- Row 3: 47 seats
From this, we can see that each subsequent row increases by 3 seats. This indicates that the number of seats in each row forms an arithmetic sequence where:
- First term (Row 1): \( a_1 = 41 \)
- Common difference: \( d = 3 \)
The formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
Where:
- \( a_n \) is the number of seats in row \( n \)
- \( a_1 \) is the number of seats in the first row
- \( d \) is the common difference
- \( n \) is the row number
Plugging in the values for row 15:
\[
a_{15} = 41 + (15-1) \cdot 3
\]
Calculating it:
\[
a_{15} = 41 + 14 \cdot 3
\]
\[
a_{15} = 41 + 42
\]
\[
a_{15} = 83
\]
So, there are **83 seats** in row 15.