30 sec approach:

It's easy to get that 1 and 3 must be true. Just consider two easiest sets:
{1, 2} the sum of even integers, which is 2, is more than the sum of odd integers, which is 1;
{2, 3} the sum of even integers, which is 2, is less than the sum of odd integers, which is 3;

Since only answer choice E offers both options (1 and 3) then it must be a correct answer. So we don't even need to consider 2.

Answer: E.

Just to elaborate on 3.
There is at least one such sequence for which Se = So: first of all, the sum of even integers is always even, hence the sum of odd integers must also be even, so # of odd terms must be even. Now, consider the set {a, a+1, a+2, a+3} (it really doesn't matter how many terms we choose, since # of odd terms is even and it really doesn't matter whether a is even or odd). In order Se = So to be true the following must be true: a+(a+2)=(a+1)+(a+3) --> 2=4, which is not true, so Se = So is not possible.

Hope it's clear.
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You quickly see that 1 & 3 are correct. As there is no 1,2,3 option, you can choose E.

Imho it is impossible for Se= So if the sequence consists only of positive integers. If you had to consider negatives too, it would be another story: -1, 0 , 1 => Se = So

Lets take a simple example:
1 2 3 4 5
The sum of all even terms is 6 and sum of all odd terms is 9

2 3 4 5 6
The sum of all even terms is 12 and sum of all odd terms is 8

x, y, x+2, y+2 .....
Sum of all even terms x + x +2 + x + 4....
Sum of all odd terms : y + y+2 + y + 4
Now for the sum to be same x = y which is not possible as they are consecutive integers. Hence the answer is 1 and 3 only
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Struggling to understand why it's a 'must be' and not a 'can be' true question. The stem does not specify the sequence, so the sequece could be anything. Option 1 sequence doesn't hold true for all Option 3 sequences, Option 3 sequence doesn't hold true for all Option 1 sequences, though the possibility is there that the sequences may be different - please can anyone clarify where I am going wrong.

Thanks

I don't think that you completely understand the question here.

The question asks: which of the following must be true?

1. There is at least one such sequence for which Se > So.

Is this statement true? Yes, there is such sequence where Se > So. For example, {1, 2}.

3. There is at least one such sequence for which Se < So.

Is this statement true? Yes, there is such sequence where Se < So. For example, {2, 3}.
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Noted, thanks for clarifying - why i got confused was because the stem said 'any sequence' and when the options say 'such sequence', the reference is obviously to the sequences mentioned in the stem, which i erroneously thought should have been true for all situations.

To get rid of statement 2: \(S(odd)=n^2\) = \(S(even)=n(n+1)\)
In this case n=0, which is not true. If you try to downgrade n +-1 step for any of parts of the equations, you will get n=-1/2, 1/3 etc which is also not true.

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