Question

1. a new movie theater is arranging rows of seats. each row farther from the screen has 4 more seats and the row in front of it. the first row (row 1) has 22 seats.

a. write a function to represent the arithmetic sequence.
b. how many seats will there be in the 14th row?
equation: $a_n = \square$
$a_{14} = \square$

Explanation:

Part a: Write the arithmetic sequence function

Step 1: Recall arithmetic sequence formula

The formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step 2: Identify \( a_1 \) and \( d \)

Here, \( a_1 = 22 \) (seats in the first row) and \( d = 4 \) (each row has 4 more seats than the previous).

Step 3: Substitute into the formula

Substitute \( a_1 = 22 \) and \( d = 4 \) into \( a_n = a_1 + (n - 1)d \):
\[

$$\begin{align*} a_n &= 22 + (n - 1) \times 4 \\ &= 22 + 4n - 4 \\ &= 4n + 18 \end{align*}$$

\]

Part b: Find the number of seats in the 14th row

Step 1: Use the arithmetic sequence function

We have the function \( a_n = 4n + 18 \). To find \( a_{14} \), substitute \( n = 14 \) into the function.

Step 2: Calculate \( a_{14} \)

\[

$$\begin{align*} a_{14} &= 4 \times 14 + 18 \\ &= 56 + 18 \\ &= 74 \end{align*}$$

\]

Answer:

• For part a, the function is \( a_n = 4n + 18 \) (or \( a_n = 22 + (n - 1) \times 4 \)).
• For part b, \( a_{14} = 74 \).

So, the equation for \( a_n \) is \( \boldsymbol{4n + 18} \) (or \( \boldsymbol{22 + 4(n - 1)} \)) and \( a_{14} = \boldsymbol{74} \).