Problem Solving

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

The OA is the option E.

How can I solve this PS question? I'm really confused here.

Experts, can you give me some help?

Hi Vincen,
Let's take a look at your question.

We will solve this question by assuming some random values for the set Z.
Let n = 5, therefore, the set Z has first 5 positive odd numbers. i.e.
$$Z=\left\{1,\ 3,\ 5,\ 7,\ 9\right\}$$

Let k = 2,
then x is the is the maximum value of the sum of k=2 distinct members of Z.
We can see that the two values of set Z, that will give the maximum sum are the last two values, i.e. 7 and 9. Therefore,
$$x=7+9=16$$

y is the is the minimum value of the sum of k=2 distinct members of Z.
We can see that the two values of set Z, that will give the minimum sum are the first two values, i.e. 1 and 3. Therefore,
$$y=1+3=4$$

Now, we can find x+y:
$$x+y=16+4=20$$

We had: n = 5 and k = 2
So 20 will be equal to:
$$2\times5\times2=2nk$$

Therefore, option E is correct.

Hope it helps.
I am available if you'd like any follow up.