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Question 1137276: an auditorium has 16 seats in the first row. 18 seats in second, 20 in the 3rd and so on through row 60. Row through 60 to 80 has the same number of seats. find the total number of seats in the auditorium
Found 2 solutions by Theo, MathLover1:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 20 to 60 forms an arithmetic sequence.
the formula for arithmetic sequence is:
An = A1 + d * (n - 1)
for example:
when A1 = 2 and n = 4 and d = 2, A4 is equal to 2 + 2 * 3 = 2 + 6 = 8.
the sequence is 2,4,6,8
the sum of an arithmetic sequence is:
Sn = n * (A1 + An) / 2.
when A1 = 2 and n = 4 and d = 2, then A4 = 6 and Sn = n * (A1 + An) / 2 becomes S4 = 4 * (2 + 8) / 2 which becomes S4 = 4 * 10 / 2 which becomes S4 = 20.
the sequence is 2,4,6,8.
the sum of the elements in the sequence is 2 + 4 + 6 + 8 = 20.
the above simply confirms that the formulas are correct.
your problem has A1 = 16 and d = 2 and n = 60.
when n = 60, An = 16 + 2 * 59 which becomes 16 + 118 = 134.
the 60th row has 134 seats in it.
the sum of all the seats from row 1 through row 60 uses the formula of Sn = n * (A1 + An) / 2 which becomes S60 = 60 * (16 + 134) / 2 which becomes S60 = 60 * 150 / 2 which becomes 30 * 150 which becomes 4500.
that gets you from row 1 to row 60.
rows 61 through 80 all have 134.
the number of rows is 80 minus 61 plus 1 = 20.
20 * 134 = 2680.
add that to 4500 and you have a total of 7180 seats in the auditorium.
your solution is that the total number of seats in the auditorium is 7180.
Answer by MathLover1(20849)  (Show Source):
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