The number of seats in the first row of an auditorium is 18

I think its D.

in the first row, we have 18 seats.
2nd = 18+2
3rd= 18 + 4, and so on
So, its 18 +2n where n starts from 0
So, total no of seats = (18+2n)n .
If, we know n , we can calculate the answer.
1. gives the value of n Sufficient
2. gives us the no of rows in the last row, so, 18+ 2n = 70, we get n, Sufficient

Hence, D.

We are given that there are 18 seats in the first row of an auditorium and that in each row after the first there are 2 more seats than in the previous row. Thus, we could say:

row 1 = 18

row 2 = 18 + 2(1) = 20

row 3 = 18 + 2(2) = 22

row 4 = 18 + 2(3) = 24

Seeing the pattern, we can set up the following expression for the number of seats in the nth row:

row n = 18 + 2(n – 1)

We need to determine the total number of seats in the auditorium. Thus, if we know the total number of rows in the auditorium, we will be able to make this determination.

Statement One Alone:

The number of rows of seats in the auditorium is 27.

Since we know that the first row has 18 seats and we know that each row following has 2 more seats than the preceding row, we could use the pattern developed above to determine the number of seats in rows 1 through 27, inclusive, and then add those values together to determine the total number of seats in the auditorium.

Note that since this is a data sufficiency question, we do not want to waste time determining the actual total number of seats. Since we know that we could determine this value, we can move on to the next statement. We eliminate answer choices B, C, and E.

Statement Two Alone:

The number of seats in the last row is 70.

We can use the equation, 18 + 2(n – 1), to determine the total number of rows.

70 = 18 + 2(n – 1)

52 = 2n – 2

54 = 2n

27 = n

Since we know that n is 27, we know that there are 27 total rows in the auditorium. We see that this is the same information we were given in statement one, and since statement one was sufficient, statement two is also sufficient, using the same reasoning.

The answer is D.

S=n/2{2a+(n-1)d} is the formula for ap as the question is in AP i.e. constantly increasing with 2 eg 18,20,22 and so on
Now, a = 1st term which is known=18
d = difference =2
1 n =27 so putting a,d,n in the first equation we can know S which is the sum of the progression.
2 number of seats in the last row is 70.
n=(final term of the series -initial term of the series)/d -1 so n can be calculated and once we know n , S can be found out .
hence answer is D

The auditorium seats are in AP. So sum of total seats shall be

Sn=n/2{ 2a+(n-1) d } , where a=18 , d=2

(1) says, n=27, so we can find the value of Sn. SUFFICIENT

(2) says, l = 70, 70 (tn) can be also written as tn=a+(n-1)d . SUFFICIENT

Both are individually sufficient. Thus answer D.

Hi All,

Certain DS questions are really just 'logic' questions - meaning that if you understand the logic involved, then you can correctly answer the question without doing much (if any) math.

Here, we're told that the 1st row in an auditorium has 18 seats and each row after has 2 more seats than the row that immediately precedes it. Thus....

2nd row = 20 seats
3rd row = 22 seats
4th row = 24 seats
Etc.

So, if you know the row number, then you'll know the number of seats AND if you know the number of seats, you'll know the row number. We're asked for the total number of rows in the auditorium.

1) The number of rows of seats in the auditorium is 27.

With this Fact, we can absolutely calculate the total number of seats (we'd just have to count them all up). Thankfully, we don't actually have to do that math to know that this is enough information to answer the given question.
Fact 1 is SUFFICIENT.

(2) The number of seats in the last row is 70.

With this Fact, we could figure out the number of seats in each of the preceding rows (68, 66, 64, etc.), so we could figure out the total number of seats. Just as in Fact 1, we don't actually have to do that math though.
Fact 2 is SUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

1) Sufficient.
2) Sufficient.

Answer: D.

Note that since this is a data sufficiency question, we do not want to waste time determining the actual total number of seats.

But if we want to find the actual total number of seats N, it can be found in multiple ways.

d= 2;
n= 27;
a1= 18;
an= a27 = 70

How we get 70?
\(an= a1+ (n-1) *d\)
a27= 18+(27-1) *2= 70


Now, total number of seats N
N=\(\frac{n*(a1+ an)}{2 }\)

N= 27*(18+70)/2= 1188


Or, from 18 to 70 is (18+ 20+ 22 …………. +70) = 2(9+ 10+11……. +35).Take out 2 common and find the sum in brackets.

We know the sum of consecutive integers \(\frac{n*(n+1)}{2}\) but only when they start from 1.
So, we find the sum of first 35 numbers (from 1 to 35) and subtract the sum of first 8 numbers from it. That will give us the sum of numbers from 18 to 70. Note that we subtract 8 numbers because 9 is part of our series.

2[35*(35+1)/2 - 8*(8+1)/2] =1188


Or, Sum = Average * Number of terms
Average= (largest term+ smallest term)/2 = (70+18)/2= 44
Numbers = (largest term - smallest term)/2+ 1= (70-18)/2 + 1= 27
Sum= 44*27= 1188

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.