Question: The number of seats in the first row of an auditorium is 18 and the number of seats in each row thereafter is 2 more than in the previous row. What is the total number of seats in the rows of the auditorium?

(1) The number of rows of seats in the auditorium is 27.
(2) The number of seats in the last row is 70.

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: D

Solution and Explanation:

Approach Solution 1:

18 seats are located in the first row of an auditorium, and 2 additional seats are located in each row after that. How many seats are there in total in the auditorium's rows?

The first row contains 18 seats. The second row contains 18+2=20. The third row contains 18+2*2=22. The fourth row contains 18+3*2=24. The number of seats in the nth row is 18+(n-1)*2.

Keep in mind that we could add the number of seats in each row to determine the total number of seats if we knew the total number of rows.

(1) There are 27 rows of seats in the theater. Sufficient.
(2) The last row has 70 seats, which is equal to 18 + (n-1)*2 = 70, where n is the number of rows. In our case, n. Sufficient.

D is the correct answer.

Approach Solution 2:

We are informed that the first row of an auditorium has 18 seats and that there are 2 additional seats in each row following the first. So, we could state:

row 1 = 18

row 2 = 18 + 2(1) = 20
row 3 = 18 + 2(2) = 22
row 4 = 18 + 2(3) = 24

We may construct the following expression for the quantity of seats in the nth row after seeing the pattern:

row n = 18 + 2(n – 1) (n – 1)

The total number of seats in the auditorium needs to be determined. Therefore, we will be able to determine this if we are aware of the total number of rows in the auditorium.

One Statement Only:

The auditorium has 27 rows of seats in total.

We could use the pattern established above to calculate the number of seats in rows 1 through 27, inclusive, and then add those values together to calculate the total number of seats in the auditorium because we know that the first row has 18 seats and that each row after it has 2 more seats than the row before.

It should be noted that since this is a question about data sufficiency, we don't want to waste time figuring out how many seats there are in total. We can go to the following sentence now that we are aware that we could find this value. B, C, and E are eliminated as possible answers.

Only Statement Two:

Seventy seats make up the final row.

To calculate the total number of rows, we can solve for 18 + 2(n - 1).

70 = 18 + 2(n – 1) (n – 1)
52 = 2n – 2
54 = 2n
27 = n

There are a total of 27 rows in the auditorium, because we know that n is 27. We can see that this is the same data that was presented to us in statement 1, and since statement 1 was adequate, statement 2 is also adequate by the same standards.

D is the correct answer.

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