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If the sum of n consecutive integers is 1, where n > 1

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If the sum of n consecutive integers is 1, where n > 1 [#permalink] New post 24 Oct 2012, 13:18
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If the sum of n consecutive integers is 1, where n > 1, which of the following must be true?

I. n is an even number
II. n is an odd number
III. The average (arithmetic mean) of the n integers is 1

(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Oct 2012, 03:16, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If the sum of n consecutive integers is 1 [#permalink] New post 24 Oct 2012, 13:43
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There's only 1 possibility...
the 2 integers are { 0, 1 }

Everything else doesn't work. If you start {-2, -1, 0, 1, 2}

you need the positive versions of the negative numbers just to get the sum equal to zero. Then the moment you add the next number in the sequence {3}, the sum goes from 0 to 3 and it skips "1" entire.y.

{-1, 0, 1} gets you to 0
{-1, 0, 1, 2} gets you to 2

The only way to get to 1 is {0, 1} and that is an even # of integers.
{1} itself doesn't work because n>1

So average of 0 and 1 is 0.5 (so III is out).
n=2 so n is an even number. There is no case where n works and is an odd number.

I only says that n is an even number and that's the only possibility.
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Re: If the sum of n consecutive integers is 1 [#permalink] New post 24 Oct 2012, 14:47
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danzig wrote:
If the sum of n consecutive integers is 1, where n > 1, which of the following must be true?
I. n is an even number
II. n is an odd number
III. The average (arithmetic mean) of the n integers is 1
(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III


Since the sum of the numbers is 1, the average of the integers is 1/n < 1, and certainly not an integer, so it cannot be 1. III is false.
Only when there is an even number of terms in a sequence of consecutive integers, the average is not an integer. Otherwise, the average is the middle term and for sure, an integer.
Therefore, I is true and II is false.

Answer A.
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Re: If the sum of n consecutive integers is 1, where n > 1 [#permalink] New post 22 Feb 2014, 10:07
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Re: If the sum of n consecutive integers is 1, where n > 1   [#permalink] 22 Feb 2014, 10:07
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