The problem in the image describes an arithmetic sequence where:

• The first row of an auditorium has 42 seats.
• Each row after the first has three more seats than the row before it.

Part A: Find the number of seats in Row 25.

In an arithmetic sequence, the general formula for the $n$-th term is:

$a_n = a_1 + (n-1) \cdot d$

Where:

• $a_1 = 42$ (seats in the first row),
• $d = 3$ (common difference, as each row has 3 more seats than the previous),
• $n = 25$ (the row number).

Now, substitute the values into the formula:

$a_{25} = 42 + (25-1) \cdot 3$ $a_{25} = 42 + 24 \cdot 3$ $a_{25} = 42 + 72 = 114$

So, there are 114 seats in Row 25.

Part B: Which row has 90 seats?

Now, we need to find $n$ when the number of seats, $a_n = 90$.

Using the same formula:

$a_n = a_1 + (n-1) \cdot d$ Substitute $a_n = 90$, $a_1 = 42$, and $d = 3$:

$90 = 42 + (n-1) \cdot 3$ Subtract 42 from both sides:

$48 = (n-1) \cdot 3$ Now, divide by 3:

$16 = n - 1$ Finally, add 1:

$n = 17$

So, Row 17 has 90 seats.

Do you have any questions or need further details? Here are some related questions to explore:

1. How do you derive the general term formula for arithmetic sequences?
2. How would the sequence change if the difference between rows was 5 instead of 3?
3. What is the total number of seats in the first 30 rows?
4. How many seats are there in Row 50?
5. What is the sum of seats in the first 17 rows?

Tip: Arithmetic sequences are characterized by a constant difference between consecutive terms. The formula helps to calculate any term directly from the first term and the common difference.