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# Z is the set of the first n positive odd numbers, where n is

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Manager
Joined: 29 Nov 2011
Posts: 80
Z is the set of the first n positive odd numbers, where n is [#permalink]

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14 Jun 2012, 04:08
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65% (hard)

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66% (03:22) correct 34% (02:40) wrong based on 212 sessions

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Z is the set of the first n positive odd numbers, where n is a positive integer. Given that n > k, where k is also a positive integer, x is the maximum value of the sum of k distinct members of Z, and y is the minimum value of the sum of k distinct members of Z, what is x + y?

(A) kn
(B) kn + k^2
(C) kn + 2k^2
(D) 2kn – k^2
(E) 2kn
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 39678
Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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14 Jun 2012, 04:21
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Smita04 wrote:
Z is the set of the first n positive odd numbers, where n is a positive integer. Given that n > k, where k is also a positive integer, x is the maximum value of the sum of k distinct members of Z, and y is the minimum value of the sum of k distinct members of Z, what is x + y?

(A) kn
(B) kn + k^2
(C) kn + 2k^2
(D) 2kn – k^2
(E) 2kn

Probably the easiest way to solve this question would be to assume some values for n and k.

Say n=3, so Z, the set of the first n positive odd numbers would be: Z={1, 3, 5};
Say k=1, so X, the maximum value of the sum of K distinct members of Z would simply be 5. Similarly, Y, the minimum value of the sum of K distinct members of Z would simply be 1.

X+Y=5+1=6.

Now, substitute n=3 and k=1 in the options provided to see which one yields 6. Only asnwer choice E fits: 2kn=2*3*1=6.

Note that for plug-in method it might happen that for some particular number(s) more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only.

Hope it helps.
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Manager
Joined: 28 Dec 2012
Posts: 111
Location: India
Concentration: Strategy, Finance
WE: Engineering (Energy and Utilities)
Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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18 Jan 2013, 01:30
for different values of n and k.. the options are invalid.
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Impossibility is a relative concept!!

Manager
Joined: 18 Oct 2011
Posts: 90
Location: United States
Concentration: Entrepreneurship, Marketing
GMAT Date: 01-30-2013
GPA: 3.3
Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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18 Jan 2013, 11:26
Just use hypothetical numbers. Say n = 5 (1,3,5,7,9) and k = 3. This would mean that x = 21 and y = 9.
x+y = 30

Only answer E gives this result.
Joined: 27 Dec 2012
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GMAT 1: 660 Q48 V33
GMAT 2: 730 Q49 V40
WE: Engineering (Energy and Utilities)
Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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19 Jan 2013, 10:03
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In the given case, sum of all n odd numbers is n^2.

If the numbers are arranged in increasing order, the max sum is when k is taken from end. So the sum of k odd numbers taken from the end
= n^2 - (n-k)^2
(Sum of all numbers) - (Sum of remaining initial numbers)
which is equal to 2nk - k^2 -- (1)

Now for minimum sum, take k numbers from the starting and their sum is k^2 -- (2)

Adding (1) and (2) = 2nk.

DJ
Intern
Joined: 17 Aug 2009
Posts: 2
Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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23 Jan 2013, 12:36
Let us name Zi the general term of the set Z.
z(i) = 2*i - 1 for i = 1 to n.
For instance z(1) = 2*1 - 1 = 1
z(2) = 2*2 - 1 = 3

z(n) = 2*n - 1

y is the sum of the first k terms of the set Z.
x is the sum of the terms from order (n-k+1) to n.

Let us express the sums x and y. Each of the sums contains k terms.
y = Sum (2*i - 1) for i = 1, ... k
x = Sum (2*j - 1) for j = (n-k+1), ,n

By developing y, we obtain
y = 2*Sum (i) - k for i=1,....k
y = 2*(1/2) [k*(k+1)]/2 - k = k^2 + k - k = k^2
(we can also remember that the sum of the first k odd numbers equals k^2).

By developing x, we obtain
x = 2*Sum (j) - k for j=n-k+1, n
We can notice that x is the difference of
- the sum of the first "n" odd numbers
- and the sum of the first (n-k) odd numbers.
We deduce
x = n^2 - (n-k)^2=2*n*k - k^2

Then x + y = 2*n*k - k^2 + k^2 = 2*n*k

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Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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09 Oct 2014, 09:08
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Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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12 Jun 2016, 00:12
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).

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Set of positive odd numbers [#permalink]

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12 Jul 2016, 09:54
Z is the set of the first n positive odd numbers, where n is a positive integer. Given that n > k, where k is also a positive integer, x is the maximum value of the sum of k distinct members of Z, and y is the minimum value of the sum of k distinct members of Z, what is x + y?

A. kn

B. kn + k^2

C. kn + 2k^2

D. 2kn – k^2

E. 2kn
Math Expert
Joined: 02 Sep 2009
Posts: 39678
Re: Z is the set of the first n positive odd numbers, where n is [#permalink]

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12 Jul 2016, 09:56
JCLEONES wrote:
Z is the set of the first n positive odd numbers, where n is a positive integer. Given that n > k, where k is also a positive integer, x is the maximum value of the sum of k distinct members of Z, and y is the minimum value of the sum of k distinct members of Z, what is x + y?

A. kn

B. kn + k^2

C. kn + 2k^2

D. 2kn – k^2

E. 2kn

Merging topics. Please refer to the discussion above.
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Re: Z is the set of the first n positive odd numbers, where n is   [#permalink] 12 Jul 2016, 09:56
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