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Manager  Joined: 31 Oct 2011
Posts: 213
For any sequence of n consecutive positive integers, Se  [#permalink]

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Difficulty:   15% (low)

Question Stats: 80% (01:34) correct 20% (01:27) wrong based on 262 sessions

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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.
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: PT #8 PS 2 Q14  [#permalink]

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eybrj2 wrote:
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.

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.

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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Manager  Joined: 21 Oct 2013
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Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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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
Manager  Joined: 20 Dec 2013
Posts: 116
Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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eybrj2 wrote:
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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Manager  B
Joined: 29 Dec 2014
Posts: 63
Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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WilDThiNg wrote:
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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Manager  B
Joined: 29 Dec 2014
Posts: 63
Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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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.
Senior Manager  G
Joined: 29 Dec 2017
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Concentration: Marketing, Technology
GMAT 1: 630 Q44 V33 GMAT 2: 690 Q47 V37 GMAT 3: 710 Q50 V37 GPA: 3.25
WE: Marketing (Telecommunications)
For any sequence of n consecutive positive integers, Se  [#permalink]

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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. For any sequence of n consecutive positive integers, Se   [#permalink] 28 Aug 2018, 08:32
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For any sequence of n consecutive positive integers, Se

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