Final answer:
The last row of the amphitheater has 491 seats, using the nth-term arithmetic sequence formula. The entire amphitheater has 30,245 seats, calculated by summing the arithmetic series.
Step-by-step explanation:
The question describes an arithmetic sequence for the seating arrangement in the amphitheater, with each row having 4 more seats than the previous one.
Number of seats in the back row:
The number of seats in each row follows the pattern 35, 39, 43, ..., with a common difference of 4. Since the front row has 35 seats and each subsequent row has 4 more seats, the last row would be the 115th term of this arithmetic sequence. To find the number of seats in the last row, we use the arithmetic sequence formula for the nth term:
nth term = a + (n - 1) × d
where a is the first term (35 seats), n is the number of terms (115 rows), and d is the common difference (4 seats). So:
115th term = 35 + (115 - 1) × 4 = 35 + 114 × 4 = 35 + 456 = 491 seats
Number of seats in the entire amphitheater:
To find the total number of seats, we sum an arithmetic series with 115 terms. The sum of an arithmetic series is given by the formula:
Sum = (n/2) × (first term + last term)
We've already determined the last term to be 491 seats and the first term is 35 seats. Thus:
Sum = (115/2) × (35 + 491) = (115/2) × 526 = 57.5 × 526 = 30245 seats