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For any sequence of n consecutive positive integers, Se denotes the sum of all even integers and So denotes the sum of all odd integers. Which of the following must be true?

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

2. There is at least one such sequence for which Se = So

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

A. 1 only
B. 2 only
C. 3 only
D. 1 & 2 only
E. 1 & 3 only.

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
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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.
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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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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.
Not really, the sequence should be consecultive integers, not necessarely starting from 1. For instance, {4,5} is a valid sequence
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