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In a certain theater, the first row has 12 seats, and each r

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In a certain theater, the first row has 12 seats, and each r

by Max@Math Revolution » Tue May 15, 2018 1:36 am

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[GMAT math practice question]

In a certain theater, the first row has 12 seats, and each row has 1 more seat than the previous row. If the last row has 50 seats, what is the total number of seats in the theater?

A. 1003
B. 1029
C. 1129
D. 1209
E. 1,339
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by GMATGuruNY » Tue May 15, 2018 3:37 am
Max@Math Revolution wrote:[GMAT math practice question]

In a certain theater, the first row has 12 seats, and each row has 1 more seat than the previous row. If the last row has 50 seats, what is the total number of seats in the theater?

A. 1003
B. 1029
C. 1129
D. 1209
E. 1,339
Since each row has 1 more seat than the preceding row, the row tallies form a set of consecutive integers:
12, 13, 14...48, 49. 50.

For any set of consecutive integers:
Count = biggest - smallest + 1
Average = (biggest + smallest)/2
Sum = (count)(average).

In the problem at hand:
Smallest = 12.
Biggest = 50.
Thus:
Count = 50 - 12 + 1 = 39.
Average = (50 + 12)/2 = 31.
Sum = 39*31 = (39)(30 + 1) = 1170 + 39 = 1209.

The correct answer is D.
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by Jeff@TargetTestPrep » Wed May 16, 2018 10:07 am
Max@Math Revolution wrote:[GMAT math practice question]

In a certain theater, the first row has 12 seats, and each row has 1 more seat than the previous row. If the last row has 50 seats, what is the total number of seats in the theater?

A. 1003
B. 1029
C. 1129
D. 1209
E. 1,339
We see that row 1 has 1 + 11 = 12 seats, row 2 has 2 + 11 = 13 seats. Say the last row is row n, and if it has 50 seats, then:

n + 11 = 50

n = 39

So the last row, row 39, has 50 seats.

Now let's determine the total number of seats in the theater:

sum = average x quantity

sum = (50 + 12)/2 x 39

sum = 31 x 39

sum = 1209

Answer: D

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by Max@Math Revolution » Thu May 17, 2018 12:27 am
=>

The question asks for the value of 12 + 13 + ... + 50. This is the sum of an arithmetic sequence with first term a = 12, and last term l = 50.
The sum of n terms of an arithmetic sequence may be found using the formula n/2 (a + l).
The number of rows is n = 50 - 12 + 1 = 39.
So, the number of seats in the theater is 39 * ( 12 + 50 ) / 2 = 39 * 62 / 2 = 39 * 31 = 1209.

Therefore, the answer is D.

Answer: D
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