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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 25 Sep 2018, 04:53
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is a positive integer. If X is the average of the odd integers in set S and Y is the average of the even integers in set S, what is the value of (X − Y)?


(A) 0

(B) 1/2

(C) 1

(D) 3/2

(E) 2
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A
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 27 Sep 2018, 17:33
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Bunuel
Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is a positive integer. If X is the average of the odd integers in set S and Y is the average of the even integers in set S, what is the value of (X − Y)?


(A) 0

(B) 1/2

(C) 1

(D) 3/2

(E) 2
Show more

In set S, there are 2n + 1 integers: n + 1 of them are odd and n of them are even. Recall that the sum of the first k positive odd integers is k^2, so the sum of the first n + 1 positive odd integers is (n + 1)^2. Likewise, the sum of first k positive even integers is k(k + 1), so the sum of the first n positive even integers is n(n + 1). Therefore, we have:

X = (1 + 3 + … + (2n + 1))/(n + 1) = (n + 1)^2/(n + 1) = n + 1

and

X = (2 + 4 + … + 2n))/n = n(n + 1)/n = n + 1

Therefore, the difference is X - Y = (n + 1) - (n + 1) = 0.

Alternate Solution:

The odd numbers in set S are 1, 3, 5, … , 2n + 1. Since these numbers form an evenly spaced sequence of integers, the average of these numbers is equal to the average of the first and last terms, which is ((2n + 1) + 1) / 2 = (2n + 2) / 2 = n + 1.

Similarly, the even numbers in set S, which are 2, 4, 6, … , 2n, also form an evenly spaced sequence of integers. Thus, the average of the even numbers in set S is (2n + 2) / 2 = n + 1.

We see that the difference is (n + 1) - (n + 1) = 0.

Second Alternate Solution:

We know that the number (2n + 1) must be odd. Therefore, the set {1, 2, 3, 4, …, 2n + 1} will begin and end with an odd integer. So let’s assume that set S = {1,2,3,4,5}. The average of the odd numbers is (1 + 3 + 5)/3 = 3. The average of the even numbers is (2 + 4)/2 = 3.

Since the two averages are equal, their difference will be 3 - 3 = 0. This difference of 0 will hold for any chosen positive integer value of n.

Answer: A
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 25 Sep 2018, 16:31
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Bunuel
Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is a positive integer. If X is the average of the odd integers in set S and Y is the average of the even integers in set S, what is the value of (X − Y)?


(A) 0

(B) 1/2

(C) 1

(D) 3/2

(E) 2
Show more
Set S ends with an odd integer, a fact we know because the last term is \(2n + 1\).

\(2n + 1\) must be an odd integer. Plug in any value to check.
\(2n\) is always Even. \(1\) is Odd.
[Even + Odd] = ODD

Let set S be {1, 2, 3, 4, 5, 6, 7}

X is the average of the FOUR odd integers:
1, 3, 5, 7
In an evenly spaced set with an even number of terms the average of the set is the average of the two middle terms:*
X = \(\frac{3+5}{2} =\frac{8}{2}=4\)
X \(=4\)

Y is the average of the THREE even integers
2, 4, 6
In an evenly spaced set with an odd number of terms, the average is the middle term:
Y \(=4\)

(X - Y) = (4 - 4) = 0

Answer
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A


*OR average is
X = \(\frac{1+3+5+7}{4}=\frac{16}{4}=4\)

Y = \(\frac{2+4+6}{3}=\frac{12}{3}=4\)
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 12 Sep 2020, 00:36
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Bunuel
Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is a positive integer. If X is the average of the odd integers in set S and Y is the average of the even integers in set S, what is the value of (X − Y)?


(A) 0

(B) 1/2

(C) 1

(D) 3/2

(E) 2

In set S, there are 2n + 1 integers: n + 1 of them are odd and n of them are even. Recall that the sum of the first k positive odd integers is k^2, so the sum of the first n + 1 positive odd integers is (n + 1)^2. Likewise, the sum of first k positive even integers is k(k + 1), so the sum of the first n positive even integers is n(n + 1). Therefore, we have:

X = (1 + 3 + … + (2n + 1))/(n + 1) = (n + 1)^2/(n + 1) = n + 1

and

X = (2 + 4 + … + 2n))/n = n(n + 1)/n = n + 1

Therefore, the difference is X - Y = (n + 1) - (n + 1) = 0.

Alternate Solution:

The odd numbers in set S are 1, 3, 5, … , 2n + 1. Since these numbers form an evenly spaced sequence of integers, the average of these numbers is equal to the average of the first and last terms, which is ((2n + 1) + 1) / 2 = (2n + 2) / 2 = n + 1.

Similarly, the even numbers in set S, which are 2, 4, 6, … , 2n, also form an evenly spaced sequence of integers. Thus, the average of the even numbers in set S is (2n + 2) / 2 = n + 1.

We see that the difference is (n + 1) - (n + 1) = 0.

Second Alternate Solution:

We know that the number (2n + 1) must be odd. Therefore, the set {1, 2, 3, 4, …, 2n + 1} will begin and end with an odd integer. So let’s assume that set S = {1,2,3,4,5}. The average of the odd numbers is (1 + 3 + 5)/3 = 3. The average of the even numbers is (2 + 4)/2 = 3.

Since the two averages are equal, their difference will be 3 - 3 = 0. This difference of 0 will hold for any chosen positive integer value of n.

Answer: A
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Hi Scott
Can you please elaborate on how you concluded that the set will have 2n+1 integers. To me (2n+1) was just an indication that the last digit if the set will be an odd number. How to conclude that - "In Set S, there are 2n + 1 integers"
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The odd numbers in the list are equally spaced. So are the even numbers. In an equally spaced list, the average and median are identical. Since the list starts and ends with an odd number, the median of the odd values is the same as the median of the even values, so the answer will be zero.

But you don't even need to think through the problem that far. Looking at the answer choices, we can be certain that this question always has the same numerical answer, so the answer to the question must be the same for every value of n. If the answer changed for different values of n, the question would have more than one right answer, which can't happen on the GMAT. So we can just answer the question when n=1, so our list is 1, 2, 3, and then the answer is clearly zero.
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 12 Sep 2020, 07:18
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S = {1, 2, 3, 4 . . . (2n + 1)}

Put n = 4 => 2(4)+1 = 9

S = {1, 2, 3, 4 ,5, 6, 7, 8, 9}

=> X = average of 1,3,5,7,9

=> \(\frac{(1 + 3 + 5 + 7 + 9 ) }{ 5}\) = \(\frac{25 }{ 5}\) = 5

=> Y = average of 2,4,6,8

=> \(\frac{(2 + 4 + 6 + 8) }{ 4 }\)= \(\frac{20 }{ 4}\) = 5

=> X - Y = 0

Answer A
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 06 Dec 2020, 10:49
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is a positive integer. If X is the average of the odd integers in set S and Y is the average of the even integers in set S, what is the value of (X − Y)?


(A) 0

(B) 1/2

(C) 1

(D) 3/2

(E) 2

In set S, there are 2n + 1 integers: n + 1 of them are odd and n of them are even. Recall that the sum of the first k positive odd integers is k^2, so the sum of the first n + 1 positive odd integers is (n + 1)^2. Likewise, the sum of first k positive even integers is k(k + 1), so the sum of the first n positive even integers is n(n + 1). Therefore, we have:

X = (1 + 3 + … + (2n + 1))/(n + 1) = (n + 1)^2/(n + 1) = n + 1

and

X = (2 + 4 + … + 2n))/n = n(n + 1)/n = n + 1

Therefore, the difference is X - Y = (n + 1) - (n + 1) = 0.

Alternate Solution:

The odd numbers in set S are 1, 3, 5, … , 2n + 1. Since these numbers form an evenly spaced sequence of integers, the average of these numbers is equal to the average of the first and last terms, which is ((2n + 1) + 1) / 2 = (2n + 2) / 2 = n + 1.

Similarly, the even numbers in set S, which are 2, 4, 6, … , 2n, also form an evenly spaced sequence of integers. Thus, the average of the even numbers in set S is (2n + 2) / 2 = n + 1.

We see that the difference is (n + 1) - (n + 1) = 0.

Second Alternate Solution:

We know that the number (2n + 1) must be odd. Therefore, the set {1, 2, 3, 4, …, 2n + 1} will begin and end with an odd integer. So let’s assume that set S = {1,2,3,4,5}. The average of the odd numbers is (1 + 3 + 5)/3 = 3. The average of the even numbers is (2 + 4)/2 = 3.

Since the two averages are equal, their difference will be 3 - 3 = 0. This difference of 0 will hold for any chosen positive integer value of n.

Answer: A


Hi Scott
Can you please elaborate on how you concluded that the set will have 2n+1 integers. To me (2n+1) was just an indication that the last digit if the set will be an odd number. How to conclude that - "In Set S, there are 2n + 1 integers"
Show more

Response:

While it’s true that the greatest integer in set S will be an odd number, we actually know more than that. We know set S contains every integer between 1 and 2n + 1.

The question describes set S as {1, 2, 3, 4, … , (2n + 1)}. The ellipsis in the description indicates that the numbers keep growing in the same fashion, i.e. every term is one greater than the preceding term. Thus, set S is actually the list of consecutive integers between 1 and 2n + 1, inclusive. That’s why there are 2n + 1 integers in this set.

Sometimes it is helpful to consider specific values in such questions to understand what’s going on. For instance, if we let n = 3, then 2n + 1 = 2(3) + 1 = 7; so set S becomes S = {1, 2, 3, 4, … , 7}. As we can see, there are 7 integers in this set.
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Set S consists of the integers {1, 2, 3, 4 . . . (2n + 1)}, where n is 19 Sep 2025, 16:06
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