if z1,z2,z3, ... zn is a series of consequtive positive

Question

ok guys, we are having short of problems. let us post some problems.......

if z1,z2,z3, ..................zn is a series of consequtive positive integers, is the sum of all the integers in this series odd?

(i) (z1+z2+z3+ ..................+zn)/n is an odd integer.
(ii) n is an odd integer.

HongHu · Answer

If n is even, then there's always even numbers of odd numbers and thus the sum would be even
If n is odd, however, depends on if the starting number is odd or even.
(i)S/n is odd
S=S/n*n if n is even, S is even; if n is odd, S is odd.
Insufficient
(II) n is odd
insufficient as it depends on the starting number

Combine together, we can determine that S is odd. Sufficient

(C)

mdf2 · Answer

per definition

odd/odd = odd
even/odd = even

odd + even = odd
odd + odd = even
finally odd + even + odd = even
as such

i. no sufficient
ii. no sufficient, we know only that n is odd

both : sufficient
as we know that the division by n (as odd) is odd
so the sum is odd

ssumitsh · Answer

(i) (z1+z2+z3+ ..................+zn)/n is an odd integer. 
(ii) n is an odd integer

I will go with A.

(i) is sufficient to prove that the sum of consecutive integers z1 to zn is odd.

HongHu · Answer

Oh my, I just realized.

S=n(n+1)/2
S/n=(n+1)/2 if it is an odd  integer , meaning n+1 is even
In other words n is odd and S is odd
It is sufficient!!!

Yes (A) is right.

ian7777 · Answer

I go for A, too. If the list had an even number of numbers, then the average would be a .5 (as in, if the list was 4,5,6,7, the average would be 5.5)

If it has an odd number of numbers, then the average is an integer, and if that integer is an odd number, then the sum must have been odd, too.

banerjeea_98 · Answer

That's a good way to figure this one. Avg of even number of +ve consecutive integers is not an integer (rather .5) and for odd number of +ve consecutive integers it is an integer.

if z1,z2,z3, ... zn is a series of consequtive positive

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