SOLUTION: An auditorium has 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on, and 30 rows of seats. How many seats are in the theatre?

Question 1197145: An auditorium has 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on, and 30 rows of seats. How many seats are in the theatre?
Found 2 solutions by math_tutor2020, Alan3354:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

This is an arithmetic sequence because we increase each term by the same amount each time.
a1 = 20 = first term
d = 4 = common difference

an = nth term
an = a1 + d(n-1)
an = 20 + 4(n-1)

Plug in n = 30 to find how many seats are in the 30th row
an = 20 + 4(n-1)
a30 = 20 + 4(30-1)
a30 = 136
There are 136 seats in the 30th row.

Now use this formula to find the sum of the first n terms of an arithmetic sequence.
Sn = sum of the first n terms
Sn = (n/2)*(firstTerm + nthTerm)
Sn = (n/2)*(a1 + an)
S30 = (30/2)*(a1 + a30)
S30 = 15*(20 + 136)
S30 = 2340
The first 30 terms of the arithmetic sequence {20,24,28,...} add to 2340.
Therefore, we have 2340 total seats.

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To verify this, you can use a spreadsheet to generate the first 30 terms.
Type 20 into cell A1. Type 24 into cell A2. Then select cells A1 and A2.

While selected, there should be a small square black box at the bottom right corner of cell A2. When you highlight your mouse over this tiny black square, the mouse cursor should turn into a plus sign.
Drag this symbol down until you have generated all 30 terms needed. You should see the sequence 20,24,28,...,132,136 for cells A1 through A30 respectively.

Alternatively, you can use the "fill series" command to generate the 30 terms needed.

In cell B1, type in =SUM(A1:A30) to quickly add up all the terms rather than manually typing them in. The result of this command is 2340.

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Answer: 2340 seats total

Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
An auditorium has 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on, and 30 rows of seats. How many seats are in the theatre?
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20 in the 1st row.
20 + 4*29 = 136 in the 30th row
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Avg = (20+136)/2 = 78
30*78 = 2340 seats total
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