If z1,....,zn is a series of consecutive positive integers, : DS Archive
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# If z1,....,zn is a series of consecutive positive integers,

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If z1,....,zn is a series of consecutive positive integers, [#permalink]

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13 Nov 2006, 13:54
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If z1,....,zn is a series of consecutive positive integers, is the sum of all the integers in this series odd?

A) (z1+....zn)/n is an odd integer

B) n is odd.
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13 Nov 2006, 14:18
If z1,....,zn is a series of consecutive positive integers, is the sum of all the integers in this series odd?

A) (z1+....zn)/n is an odd integer

B) n is odd.

FROM ONE

EVEN/EVEN = ODD OR ODD/ODD = ODD ...NOT SUFF

FROM B

N IS ODD THUS SUM IS EITHER EVEN OR ODD.........NOT SUFF

BOTH TOGETHER

SUM HAS TO BE ODD ....SUFF

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Re: DS: series of consecutive integers [#permalink]

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13 Nov 2006, 17:35
Hmm..I get A..

sum of all positive consecutive integers.. well I cant come up with a sum of consecutive numbers whose sum is odd and the number of these elements is even..

e.g. 1+2+3+4+5 = 15/5 => 3 which is odd...

netcaesar wrote:
If z1,....,zn is a series of consecutive positive integers, is the sum of all the integers in this series odd?

A) (z1+....zn)/n is an odd integer

B) n is odd.
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13 Nov 2006, 18:52
St2 first --> this tells us we have odd number of consecutive integers. If the set is {1,2,3,4,5} we have odd sum. If the set is {2,3,4,5,6}, we have an even sum. It depends if the first integer in the series is even or odd. Insufficient.

St1:

(z1+....zn)/n = odd
If the set is {1,2,3,4,5}, then the division is odd.

If the set is {2,3,4,5,6}, then the division is even.

Testing another set where starting number is even {10,11,12,13,14}, the division is even.

So since the resulting division is odd, then we have an odd number of integers where the starting number in the series is odd, causing the resulting sum of the series to be odd.

Sufficient.

Ans A
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13 Nov 2006, 18:57
I got A for this one.
From Stem: In consecutive numbers, the average of an odd number of consecutive integer will always be an integer. The average of an even number of integer will NEVER be an integer because it will be the average of the 2 middle number

For example: the average of 1+2+3+4 will be 2.5 and the average of 1+2+3+4+5 will be the middle number 3.

From (1): Since the average is an odd integer, we will also know that there are an odd number of consecutive integer. So, Sum = Odd *odd = Odd. Suff

From (2): Just because n is odd, sum could either be odd or even. For example sum of 1+2+3+4+5 = 15 = odd when there are 5 numbers. For 1+2+3+4+5+6+7 = 28 = even when there are 7 numbers.
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13 Nov 2006, 19:42
great explaination by enola......
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13 Nov 2006, 19:51
Average of 'n' consecutive numbers is an integer when n is odd.

If sum/n= odd and n=odd, then sum=odd.

Hence A.
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14 Nov 2006, 09:58
You are right.

OA is A.
14 Nov 2006, 09:58
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