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Question 953582: An amphitheater has 22 seats in the first row and 40 rows in all. Each successive row has one additional seat. How many seats are in the amphitheater?
I believe 1660 seats but I did a long explanation of this and don't know a way to find it easier I made a list adding one to every number for 40 rows excluding the first one.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe this is an arithmetic progression.
the formula for the sum of an arithmetic progression is:
Sn = n*(A1+An)/2
For example:
If you have the arithmetic progression 1,2,3,4,5, then the sum is 15.
The formula would be:
Sn = 5 * (1 + 5)/2 = 5 * 6/2 = 5 * 3 = 15.
The formula will get you the sum without all the tears.
Sometimes you have to find An first.
The formula for that is:
An = A1 + (n-1) * d
d is the common difference.
In case you didn't figure it out by now:
A1 is the first term in the series.
An is the nth term in the series.
n is the number of terms in the series.
Let's apply these formulas to your problem:
Your first row is 22 seats, so A1 = 22.
Each succeeding row has 1 more seat, so d = 1.
There are 40 rows in all, so n = 40
First we have to find An.
Then we can find Sn.
An = A1 + (n-1) * d.
replace A1 with 22 n with 40 and d with 1 and the formula becomes:
An = 22 + 39 * 1 = 22 + 39 = 61.
Sn = n * (A1 + An) / 2.
replace n with 40 and A1 with 22 and An with 61 and the formula becomes:
Sn = 40 * (22 + 61) / 2 = 40 * 83/2 = 40 * 41.5 = 1660
You got the right answer.
The sum is 1660.
The formulas, if you remember them, make the job a lot easiwer.
The formulas are, once again:
An = A1 * (n-1)*d
Sn = n * (A1 + An) / 2
Here's some good links for future reference:
http://www.regentsprep.org/regents/math/algtrig/atp2/arithseq.htm
http://www.regentsprep.org/regents/math/algtrig/atp2/geoseq.htm
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