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

for different values of n and k.. the options are invalid.

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.

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

The answer is E.

[quote="Bunuel"]
You mentioned that Say k=1, so X, the maximum value of the sum of K distinct members of Z would simply be 5.

but for the example that you have used - Z={1, 3, 5} - shouldn't the maximum value of the sum of K distinct members of Z be = 1+3+5 = 9 ?

Please help were am I going wrong ?

hi, He assumed K = One distinct member ; so max sum of One distinct member of Z going to be 5 and minimum is going to be 1.

We can let n = 5 and k = 3. Thus Z = {1, 3, 5, 7, 9}, x = 5 + 7 + 9 = 21, y = 1 + 3 + 5 = 9 and hence x + y = 21 + 9 = 30.

Now let’s check the given answer choices (note: we will be looking for the one that is equal to 30):

A) kn = 5(3) = 15 → This is not 30.

B) kn + k^2 = 5(3) + 3^2 = 15 + 9 = 24 → This is not 30.

C) kn + 2k^2 = 5(3) + 2(3)^2 = 15 + 18 = 33 → This is not 30.

D) 2kn - k^2 = 2(5)(3) - 3^2 = 30 - 9 = 21 → This is not 30.

E) 2kn = 2(5)(3) = 30 → This is 30

Answer: E

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