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​?

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Part 1
There are
  
enter your response here seats in row 15.

1 answer

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.