For any sequence of n consecutive positive integers, Se

Question

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.

Bunuel · Accepted Answer

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.

unceldolan · Answer

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

PerfectScores · Answer

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

WilDThiNg · Answer

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

Thanks

Bunuel · Answer

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}.

WilDThiNg · Answer

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.

Hero8888 · Answer

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.

mmdfl · Answer

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

For any sequence of n consecutive positive integers, Se

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

For any sequence of n consecutive positive integers, Se

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards