DS: Sum of the terms in the sequence

[zloebelf]

Problem:
Sequence a1,a2,a3,...an of n integers is such that ak = k if k is odd and ak = ak-1if k is even. Is the sum of the terms in the sequence positive?

1) n is odd
2) an is positive

My Interpretation:
k ak
1 1
2 -1
3 3
4 -3
5 5
6 -5
etc

A was the answer I provided. I understand why A is correct, but at least according to my understanding of the sequence, if an is positive, then the sum of the terms is either a positive or zero. Is this sufficient to answer than the sum is positive, even if zero is possible? Isn't zero neither positive nor negative?

[tim]

To answer your immediate question, 0 is neither positive nor negative. There is a problem with the way you have written the question though, as it appears that the sequence will be as follows:

a1 = 1
a2 = 1
a3 = 3
a4 = 3
a5 = 5
a6 = 5
etc.

If this is true, statement 1 works, but so does statement 2. More importantly though, given the problem as you wrote it, the question is answerable without either statement. This NEVER happens on the GMAT. Can you double check your transcription of the problem?

[abhishekharitwal]

Dear Sir,

Cant the above series be of negative numbers?

A1=-1
A2=1
A3=-3
A4=3.. and so on?

In that case statement 1 alone will not be sufficient.

Please guide.
Thanks.

[tim]

i need to know that the problem has been transcribed correctly before i will be able to provide any further insight into this one..

[TobiasF629]

Hi, I struggle with the same question. Here the correct question:

The Sequence a(1), a(2),a(3),..., a(n) of n integers is such that a(k) = k if k is odd and a(k) = -a(k-1) if k is even. Is the sum of the terms in the sequence positive?

(1) n is odd
(2) a(n) is positive

The brackets are supposed to indicate indices.

The correct answer is D, but in my opinion C should be the right answer. HELP!

[tim]

Given your version of the problem, statements 1 and 2 are equivalent. In other words, a(n) is positive if and only if n is odd (see if you can figure out why). Whenever this happens, D and E are the only possible answers. That alone rules out C, but my analysis above can be applied to this version of the problem to demonstrate why statement 1 is sufficient, and of course if the statements are equivalent that means statement 2 is sufficient as well.

I also want you to consider more generally the nature of the mistake you made: if you arrived at C without being able to come up with two different answers to the question at the top of the page with statement 1 and then with statement 2, then you did not do enough work to be sure that the statements were insufficient. Did you *actually* come up with two different answers to the question? In other words, were you able to take the fact that n is odd and come up with a positive value *and* a nonpositive value for the sum? If you did, then you made a calculation error somewhere and need to be more careful. If you did not, then your approach to data sufficiency is fundamentally flawed and needs some serious work.

[TobiasF629]

Hi Tim, I think that I found my mistake: I assumed that k could also be negative. In this case statement 1 would get me two scenarios. Thank you for your advice!

[RonPurewal]

Hi Tim, I think that I found my mistake: I assumed that k could also be negative.

just for the record, that is addressed here:

Sequence a1,a2,a3,...an of n integers

[RonPurewal]

also, more generally and more importantly, EVERY sequence of 'a sub n' on the gmat will ALWAYS start with 'a sub 1'.
literally, every single one, every time, all the time.

mostly this is just a case of the more general principle i explained here:
https://www.manhattanprep.com/gmat/foru ... ml#p116154
basically, it would be 'tricky' to start a sequence with any thing other than 'the FIRST thing'... 'cause... y'know... that's what the word 'first' means. (:
(at least to non-mathematicians, anyway)

in physics and calculus a large number of important sequences start with the 0th term rather than the 1st term. but they are not going to do that here.

[DeekshaB1]

Hi Ron,

For this question, isn't it possible that 'a' can be positive or negative. This will give two sequences:
1) a, -a, 3a, -3a OR
2) -a, a, -3a, 3a

Based on this I thought the answer would be C. Please help.

[RonPurewal]

look at the definition more carefully. you can't start the sequence with an arbitrary number.

the sequence is defined as a(k) = k if k is odd.
this means that
a(1) = 1
a(3) = 3
a(5) = 5

you can't make these any other numbers. (...and thus the values of the even-numbered terms are fixed, too, because those are just the negatives of these.)