The sum of first N-1 terms of AP is either zero or positive. but sum

The sum of first N-1 terms of AP is either zero or positive. but sum of first N terms is negative. What is the value of N?

A) Common difference is -4 and 7th term is last positive term
B) 25 is first term and there are 7 positive terms and all are integers

In an arithmetic sequence, if we know one term and we know the common difference, we know everything else. So we can rephrase this question: what is one term in the sequence and the common difference?

(1) Just by looking at the statement, you might guess this is insufficient because although we have the common difference, we don't have any terms in the sequence. Let's process this a bit just to make sure.

If the common difference is -4, and the 7th term is the last positive term, that means that the 7th term is either 1, 2, 3 or 4. Because the common difference is -4, the first term with be (term 7) + 24.

If the 7th term is 4,
- the 8th term is 0,
- the 9th through 15th terms all cancel out the 1st through 7th terms.
- the 16th term is when the sum becomes negative.
In this case, N must be 16.

However, if the 7th term is 2,
- the 8th term is -2
- so the 8th through 14th terms cancel out the 1st through 7th terms.
- the 15th term is when the sum becomes negative.
In this case, N must be 15.

(If you work it out for the others, N is 15 when the 7th term is 3, and N is 14 when the 7th term is 1.)

So (1) is insufficient.

(2) If there are 7 positive terms and they are all integers, then the common difference must be -4 (it cannot be -3 or -5). Since we know the first term and we know the common difference, then we can conclude that this statement is sufficient.