A movie theatre has 30 rows. The first row has 26 seats and as you move to the back each additional row has one seat added. How many seats are in the theatre?
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Number of seats in first row = 26
Number of seats in last row = 26+30-1=55
We can find the sum using Gauss's method:
26+27+28+.....+54+55=x
55+54+.....+28+27+26=x
Add the two rows
81+81+.....81 (30 rows) = 2x
30*81 = 2x
x=1215 seats
To find the total number of seats in the movie theatre, we can calculate the sum of the seats in each row.
First, let's find the number of seats in each row as you move to the back. We know that the first row has 26 seats, and each subsequent row adds one seat. This means that the second row will have 26 + 1 = 27 seats, the third row will have 26 + 2 = 28 seats, and so on.
Now we can find the sum of the seats in each row. We know that there are 30 rows in total. To find the sum, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term a is 26 (the seats in the first row) and the last term l can be calculated by adding the number of rows (n) minus 1 to the first term a. So the last term l = a + (n - 1).
Using these values in the formula, we can calculate the sum:
Sn = (n/2)(a + l)
= (30/2)(26 + (26 + (30 - 1 - 1)))
= (30/2)(26 + (26 + 28))
= (30/2)(26 + 54)
= 15(80)
= 1200
Therefore, there are a total of 1200 seats in the movie theatre.