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

A is not self sufficeient as starting term of AP can have multiple values
B looks sufficient as knowledge of -4 as common diff can be determined easily as they have 7 positive terms including 25 . also it is sure that difference should be negative to satisfy the condition of question.

So 6 numbers with 25 difference means value of -1,-2,-3 or -4 for difference , if you check all these, only -4 can satisy the condition of sums of N-1 terms as positive or 0 and N terms as negative.

Please let me know if answer and reasonign is correct.

First, I don't think the subject matches the question.

Now onto the question. I think your reasoning is generally right. Let me expand on it. Since the values are relatively small, we can test concrete numbers.

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.

_________________
Ben Ku
Instructor
ManhattanGMAT

I have a question here :

In Option B : It says there are 7 positive integers, does it mean that there are only 7 positive integers. Do we need explicit "only" to conclude there are no more than 7. or this statement is safe to conclude there are no more tha 7 positive integers.

before we comment any further - what is the source of this question?

i have serious doubts that this is from the GMATPREP SOFTWARE (this folder is specifically dedicated to problems from that software). first of all, an official problem would NEVER specify three terms in one statement (as in statement 2 here). second, the gmat won't require you to understand terms like "arithmetic progression" at all, let alone abbreviate them as "AP".

so:
* if this IS from the gmatprep software, attach a screen shot please.
* if it's NOT from the gmatprep software, please re-post it, along with your questions, in the GENERAL MATH FOLDER.

thanks.