To solve the problem, we analyze it as an arithmetic sequence. Let's break it into steps.

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Step 1: Find the number of seats in row 53.

The number of seats in the rows forms an arithmetic sequence:

• The first term ($a_1$) = 80 (seats in the first row).
• The common difference ($d$) = 5 (each row has 5 more seats than the previous one).
• The general term of an arithmetic sequence is given by: $a_n = a_1 + (n - 1)d$ where $a_n$ is the number of seats in the $n$-th row.

Substitute $n = 53$, $a_1 = 80$, and $d = 5$: $a_{53} = 80 + (53 - 1)(5)$ $a_{53} = 80 + 52 \times 5$ $a_{53} = 80 + 260 = 340$

So, the 53rd row has 340 seats.

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Step 2: Find the total number of seats in the auditorium.

The total number of seats is the sum of the arithmetic sequence: $S_n = \frac{n}{2} \times (a_1 + a_n)$ where:

• $n = 56$ (total rows),
• $a_1 = 80$ (seats in the first row),
• $a_n = a_{56} = 80 + (56 - 1)(5)$.

First, calculate $a_{56}$: $a_{56} = 80 + (56 - 1)(5) = 80 + 55 \times 5 = 80 + 275 = 355$

Now calculate the total number of seats: $S_{56} = \frac{56}{2} \times (80 + 355)$ $S_{56} = 28 \times 435 = 12,180$

So, the total number of seats in the auditorium is 12,180.

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Final Answers:

1. The 53rd row has 340 seats.
2. The auditorium has 12,180 seats.

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Do you want more details or have any questions?

Here are some related questions to explore further:

1. How many seats are in the last row of the auditorium?
2. How many additional seats are added with each new row in the first 10 rows?
3. What is the average number of seats per row in the auditorium?
4. If a new section with 10 rows is added with 5 additional seats per row, how many seats would this add?
5. What would be the total number of seats if there were only 30 rows?

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Tip: For arithmetic sequences, always check the difference between terms ($d$) to verify it is constant. This ensures the sequence formula applies correctly.