A theater has 30 rows of seats.The first row contains 20 seats,the second row contains 21 seats,and so on,each row has one more seats than the previous one.How many seats are there in the theater?
S30 = 30/2 (20 + 29*1)
To find the total number of seats in the theater, we can use the formula for the sum of an arithmetic series.
First, let's calculate the number of seats in each row using the given pattern. We know that the first row has 20 seats, and each subsequent row has one more seat than the previous row. So we can create a sequence of number of seats in each row as follows:
20, 21, 22, 23, ...
Since the number of seats is increasing by 1 with each row, we can observe that this is an arithmetic sequence where the first term (a) is 20 and the common difference (d) is 1.
Next, we need to find the number of rows in the theater. We are given that there are 30 rows in total.
Now, we can use the formula for the sum of an arithmetic series to calculate the total number of seats:
Sn = n/2 * (2a + (n-1) * d)
Where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.
Substituting in the given values:
n = 30 (number of rows)
a = 20 (number of seats in the first row)
d = 1 (number of seats increasing per row)
Sn = 30/2 * (2 * 20 + (30-1) * 1)
Simplifying, we have:
Sn = 15 * (40 + 29)
Sn = 15 * 69
Sn = 1035
Therefore, there are 1035 seats in the theater.