An auditorium has 150 rows of seats. The first row has 15 seats, the second row has 17 seats, the third row has 19 seats, and so on, each row having two more seats than the previous row. How many seats are there together in the auditorium.
S150 = 150/2 (2*15 + 149*2)
150 / 2 ( 2 x 15 + 149 x 2) what does that equal?
24600
To find the total number of seats in the auditorium, we need to sum up the number of seats in each row. We are given that the first row has 15 seats and each subsequent row has two more seats than the previous one.
To find the number of seats in each row, we can use the formula: Number of Seats = First Row Seats + (Row Number - 1) * Seat Difference
In this case, the first row has 15 seats and the seat difference is 2. So we have:
Number of Seats = 15 + (Row Number - 1) * 2
Now, we can find the number of seats in each row by substituting the row number into the formula. We need to calculate this for all 150 rows and add up the results.
Let's do the calculation for a few rows to illustrate:
For the first row (Row Number = 1):
Number of Seats = 15 + (1 - 1) * 2 = 15
For the second row (Row Number = 2):
Number of Seats = 15 + (2 - 1) * 2 = 15 + 2 = 17
For the third row (Row Number = 3):
Number of Seats = 15 + (3 - 1) * 2 = 15 + 4 = 19
We can see that the number of seats increases by 2 for each subsequent row. We can continue this calculation for all 150 rows, but there is a shortcut to find the total number of seats.
Since the number of seats in each row increases by 2, we can consider this as an arithmetic series with the first term (a) being 15 and the common difference (d) being 2. The formula to find the sum of an arithmetic series is:
Sum = (n/2) * (2a + (n-1) * d)
Here, n is the number of terms, which is the number of rows in this case (150). So, we can substitute the values into the formula to find the total number of seats:
Sum = (150/2) * (2*15 + (150-1) * 2)
= 75 * (30 + 298)
= 75 * 328
= 24,600
Therefore, there are 24,600 seats together in the auditorium.