Question

A theater has 22 rows of seats. if there are 27 seats in the first row, 30 in the 2nd, 33 in the third, and so on. how many seats are there in all?

Answer 1

To find the total number of seats in the theater, we use the sum formula of an arithmetic sequence. We calculate that the 22nd row has 90 seats, leading to a total of 1287 seats in the theater.

Step-by-step explanation:

To determine the total number of seats in the theater, we recognize that this is an arithmetic sequence problem. The number of seats in each row forms an arithmetic sequence because there is a common difference between the number of seats in subsequent rows. Starting with 27 seats in the first row and increasing by 3 seats per row, we can use the formula for the sum of an arithmetic series.

The formula for the sum of the first n terms of an arithmetic sequence is: Sn = n/2 × (a1 + an), where:

• n is the number of terms in the sequence,
• a1 is the first term of the sequence,
• an is the last term of the sequence.

To find an, the number of seats in the last (22nd) row, we use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1) × d, where d is the common difference. Here, d = 3, as the number of seats increases by 3 each row.

a22 = 27 + (22 - 1) × 3

a22 = 27 + 21 × 3

a22 = 27 + 63 = 90

Now, we can calculate the total number of seats (Sn):

S22 = 22/2 × (27 + 90)

S22 = 11 × 117 = 1287

Therefore, the theater has a total of 1287 seats.