Sequence of integers and arithmetic progressions

[hudson]

given any two numbers k,j
what is the largest sequence of integers such that the sum of any k consecutive terms is negative and the sum of any j consecutive terms is positive
and
how may we find a subset containing k of the first n numbers such that out of all subsets with k elements, this subset has the smallest arithmetic progression with common difference j. and out of all subsets with k elements, does this subset contain the largest arithmetic progression for some other integer?

ideas would be very much appreciated

 

[Eynstone]

how may we find a subset containing k of the first n numbers such that out of all subsets with k elements, this subset has the smallest arithmetic progression with common difference j.

Partition {1,..n} into j arithmetic progressions with common difference j. Choose one (if k<j) or {k/j} (if k>j) numbers from each partition.

 

[chaoseverlasting]

I don't know about the terminology required or if this is even a shot in the right direction, but for the sum of a sequence of k integers to be negative, [k/2] +1 integers must lie to the left of origin and [k/2] to the right.

Assuming j>k, for the sum of sequence to be positive, [k/2]+1 must lie to the right of the origin, and again, [k/2] to the left.

As the question says that for any sequence of k integers amongst a set, the sum should be negative and in the same set, the sum of j integers must be positive.

Let us assume a set of 'n' integers. If n is odd, and the set centered at 0, then it ranges from (-[n/2],[n/2]). In this case sum for k=n will be zero. For k<n, there will always be a set of k integers which has positive sum. Thus, the set cannot be centered at zero.

For the set centered at any point to the left of zero (say -1), the maximum value for k can be k<=n-1, for which the sum of any k integers is zero. For the set centered at any point -p ( p to the left of zero), the maximum value of k<=n-p.

However, in this case, there is no value of j which satisfies the required condition, because there will always be p cases where the sum of integers is less than zero.

Idk, it seems like there is no possible set in which consecutive integers can simultaneously satisfy both conditions. Again, I'm not familiar with any branch of number theory, just my 2 cents worth. Are there any other conditions on the sequence of integers that we're missing (perhaps that the sequence is an ap or a gp or something) ?