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# For any sequence of n consecutive positive integers, Se

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

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24 Feb 2012, 23:02
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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.
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Posts: 58445
Re: PT #8 PS 2 Q14  [#permalink]

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25 Feb 2012, 00:16
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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Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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16 Jul 2014, 04:12
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
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Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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16 Jul 2014, 05:52
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
Joined: 29 Dec 2014
Posts: 63
Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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09 Mar 2017, 05:56
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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Posts: 58445
Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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09 Mar 2017, 06:08
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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Re: For any sequence of n consecutive positive integers, Se  [#permalink]

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09 Mar 2017, 06:42
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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For any sequence of n consecutive positive integers, Se  [#permalink]

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28 Aug 2018, 08:32
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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