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# If the sum of n consecutive integers is 0, which of the following must

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Joined: 02 Sep 2009
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If the sum of n consecutive integers is 0, which of the following must  [#permalink]

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18 Sep 2018, 00:48
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71% (00:52) correct 29% (00:49) wrong based on 90 sessions

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If the sum of n consecutive integers is 0, 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 0

A. I only
B. II only
C. III only
D. I and III
E. II and III

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Re: If the sum of n consecutive integers is 0, which of the following must  [#permalink]

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18 Sep 2018, 01:04
Bunuel wrote:
If the sum of n consecutive integers is 0, 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 0

A. I only
B. II only
C. III only
D. I and III
E. II and III

{-2,-1,0,1,2}
i.e. Number of terms in the set must be ODD

For terms in Arithmetic Progression
Mean of terms = median of terms = Average of First and Last term

Which n this case will be zero hence

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Re: If the sum of n consecutive integers is 0, which of the following must  [#permalink]

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18 Sep 2018, 01:18
Let the consecutive no. as mentioned below:
-2,-1,0,1 and 2 - sum is zero.
statement I: N cannot be even. N is always odd.In above N=5.
Statement II: N is always odd.
STATEMENT III: average above is Zero.
Correct answer is both II and III.

Corrct Option is E

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If the sum of n consecutive integers is 0, which of the following must  [#permalink]

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18 Sep 2018, 01:56

Solution

Given:
• Sum of n consecutive integers = 0

To find:
• Which of the given options is always true?

Approach and Working:
• Let the ‘n’ consecutive terms be {a, a+1, a+2, a+3, …., a+n-1}, where a is an integer.
• Sum of all these n consecutive terms = $$a*n + (1+2+3+ … + n-1) = an + (n-1) * \frac{(n-1+1)}{2} = an + \frac{n(n-1)}{2}$$
o Given this sum = 0
o That is, $$an + \frac{n(n-1)}{2} = 0$$
o Implies, $$a + \frac{(n-1)}{2} = 0$$
o Which gives, n = -2a +1 …………... (1)
• Thus, n = 2p + 1, where p = -a = integer (since, a is an integer)

Therefore, n is an odd number

• Average of n consecutive integers = $$\frac{(first term + last term)}{2} = \frac{(a + a+n-1)}{2}$$
• Substituting the value of n from equation (1), we get,
o Average of n terms = $$\frac{[2a + (-2a +1) -1]}{2} = 0$$

Therefore, statements II and III must be true.

Hence, the correct answer is option E.

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Re: If the sum of n consecutive integers is 0, which of the following must  [#permalink]

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23 Sep 2018, 16:34
Bunuel wrote:
If the sum of n consecutive integers is 0, 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 0

A. I only
B. II only
C. III only
D. I and III
E. II and III

Since the sum of n consecutive integers is zero, we need zero as part of the sum , as well as an equal number of positive and negative numbers; thus, n must be odd.

Additionally, if the sum of n integers is zero, then zero divided by any nonzero number is zero, so the average is zero.

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Re: If the sum of n consecutive integers is 0, which of the following must &nbs [#permalink] 23 Sep 2018, 16:34
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