Given a series of n consecutive positive integers, where n >

Responding to a pm.

Couple of things:
In any evenly spaced set (AP) the arithmetic mean (average) is equal to the median and can be calculated by the formula: (first term+last term)/2

Now, set of consecutive integers is an evenly spaced set (AP with common difference of 1) and in order mean=median to be an integer it has to have an odd number of terms. If there are an even number of terms mean=median will be integer/2.

For example:
{1, 2, 3} --> mean=median (middle term)=(3+1)/2=2;
{1, 2, 3, 4} --> mean=median=(1+4)/2=5/2=2.5.

Next, in AP if the first term is \(a\) and the common difference of successive members is \(d\), then the \(n_{th}\) term of the sequence is given by: \(a_ n=a+d(n-1)\). In case of consecutive integers (so when common difference=d=1) the formula becomes: \(a_ n=a+n-1\).

Check Number Theory for more on AP: math-number-theory-88376.html (specifically "Consecutive Integers" and "Evenly Spaced Set" chapters of it)

BACK TO THE ORIGINAL QUESTION.
Given a series of n consecutive positive integers, where n > 1, is the average value of this series an integer divisible by 3?

Basically we are asked whether: \(average=\frac{first \ term+last \ term}{2}\) is divisible by 3 or whether \(average=\frac{a+(a+n-1)}{2}=a+\frac{n-1}{2}\) is divisible by 3 (where \(a\) is the first term and \((a+n-1)\) is the \(n_{th}\), so last term).

(1) n is odd --> as n=odd then the average is definitely an integer, though it may or may not be divisible by 3: {2, 3, 4} - YES, {1, 2, 3} - NO. Not sufficient.

(2) The sum of the first number of the series and (n – 1) / 2 is an integer divisible by 3 --> we are directly given that \(a+\frac{n-1}{2}\) is divisible by 3. Sufficient.

Answer: B.

Important note: you should have spotted that there was something wrong with your solution as on the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other.

So we can not have answer NO from statement (1) and answer YES from statement (2) (as you got in your solution), because in this case statements would contradict each other.

Hope it helps.