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# If Z_1 , Z_2 , Z_3 , ..., Z_n is a sequence of consecutive

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Manager
Joined: 07 Apr 2009
Posts: 144

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14 Jul 2009, 10:54
Quote:
If the average of a set of consecutive integers is an integer, then the total number of integers in the set will always be odd.

ie, if there are 5 consecutive integers, only then their average will be an integer, if there were 4 integers, the avg will be a decimal.

Now, the average of consecutive integers will always have equal number of integers below it as above it. Thus, say A is the avg, and A has x number of integers below it, and x number of integers above it.

so total number of integers in the set = A + 2x

Now Even + odd = odd
So if A is odd, adding it to 2x will always give an odd integer, ie, the sum of all the integers will be odd.

coming back to understanding your logic

1) I understood that if the avg of n consequtive numbers is an integer then the average should be odd
2) and i agree that there will be equal number of integers on both sides of the average(x on each side)
3) but in your explanation you said "So if A is odd, adding it to 2x will always give an odd integer, ie, the sum of all the integers will be odd."

I thought A+ 2x is just number of integers but you seem to conclude it as Sum of the numbers, irrespective of whether either side of 'A' has any number of evens or odds.

I got the point using numbers but just want to follow your logic through the end
Senior Manager
Joined: 04 Jun 2008
Posts: 279

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14 Jul 2009, 11:43
skpMatcha wrote:

coming back to understanding your logic

1) I understood that if the avg of n consequtive numbers is an integer then the average(not avg, N) should be odd
2) and i agree that there will be equal number of integers on both sides of the average(x on each side)only when they are consecutive; for other cases, there is no guarantee
3) but in your explanation you said "So if A is odd, adding it to 2x will always give an odd integer, ie, the sum of all the integers will be odd."

I thought A+ 2x is just number of integers but you seem to conclude it as Sum of the numbers, irrespective of whether either side of 'A' has any number of evens or odds.
all of the x integers below A will be of the same nature as those of above A, ie if the sum of the x integers below A is odd, so will be the sum of integers above A (simply because they are consecutive). Similarly if the sum of lower x is even, sum of greater x will be even. In either of the cases, the total sum of lesser and greater set will always be even. That is what i meant by 2x. Sorry for using short cuts.

I got the point using numbers but just want to follow your logic through the end

Its good to know that you want to get it crystal clear, but im afraid you might generalize the rule unless you practice it with different scenarios, and also, you seem to be mistaken in a few places.
Manager
Joined: 07 Apr 2009
Posts: 144

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14 Jul 2009, 11:49
Got it ! Thanks for the explanation. I am a newbie and am afraid that I might misunderstand .. I am learning so much from these forums.

Thanks again !

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Re: DS -No. Prop   [#permalink] 14 Jul 2009, 11:49

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