In any sequence of n nonzero numbers, a pair of consecutive terms with

s(k) = (-1)^k

We see that s(k) is the sequence:

-1, 1, -1, 1. ... , (-1)^n

However, since we don’t know the value of n, we can’t determine the number of sign changes or whether it’s even. For example,

If n = 2, then:

s(1) = -1, and s(2) = 1, so we have an odd number of sign changes.

If n = 3, then:

s(1) = -1, s(2) = 1, and s(3) = -1 so we have an even number of sign changes.

Statement one alone is not sufficient.

Statement Two Alone:

n is odd

WIthout knowing the sequence, we do not have enough information to determine an answer. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

As we’ve shown in statement one, when n is 3, we see that we have an even number of sign changes. When n = 5, the sequence s(k) is: -1, 1, -1, 1, -1, which have 4 sign changes. If we increase n from 5 to 7, we will have 2 more sign changes, therefore, we can see that if n is odd, the sequence s(k) = (-1)^k will always produce an even number of sign changes.

Answer: C
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Instead of reasoning about the sequence, we'll try simple numbers to give contradictory answers.
This is an Alternative approach.

(1) If n = 2 then the sequence is -1, 1 and we have 1 sign change. If n = 3 then the sequence is -1, 1, -1 and we have 2 sign changes. Insufficient.
(2) With no information on the sequence we can build whatever we like... insufficient.
Combined: If n = 3 we saw 2 sign changes. If n = 5 we have -1, 1, -1, 1, -1 which takes the same 2 sign changes from before and adds another 2. Similarly, each consecutive odd number will add another 2 sign changes so we will always have an even number of changes in total.

(C) is our answer.

Hi jabhatta@umail.iu.edu,
As also explained above by EMPOWERgmatRichC, The example is given just to help you understand the term -a pair of consecutive terms with opposite signs represents a sign change. Now the consecutive term used is for the placement of elements in the sequence, that is a,x,b.. a and x are consecutive terms, so also x and b. They are not consecutive integers.
Rather -2, 3, -4, 5...are not consecutive integers. They are 2 patterns - even terms(3,5..) and odd terms(-2,-4)

So statement II is NOT sufficient.
However statement tells us that alternate terms are of same sign...
(1) \(s_k=(–1)^k \) for all positive integers k from 1 to n...So -1, 1, -1....
Thus change in sign will be n-1. If n is odd, n-1=even and vice versa.

Combined, we know that n is odd, so change in sign = n-1=even
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Let \(x\) be the number of sign changes of the sequence. The original question: Is \(x=E\) ?

1) The consecutive terms of the sequence defined in 1) will have opposite signs, so \(x=n-1\). If \(n=E\), then \(x=E-O=O\), so the answer to the original question is No. However, if \(n=O\), then \(x=O-O=E\), so the answer to the original question is Yes. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

2) We know that the number of terms of the sequence is odd, but we don't know anything about the rule defining the consecutive terms. \(\implies\) Insufficient

1&2) We know that \(n=O\), so \(x=O-O=E\). Thus, the answer to the original question is a definite Yes. \(\implies\) Sufficient

Answer: C

I dont understand why the knowledge sequence is required isnt total no. of term sufficient to calculate answer? pls explain with example if any.

The key here is that we don’t know the total number of the terms in the sequence. It just says there are n terms in the sequence, but we don’t know the value of n. Now even if we knew the value of n, we still would not know the number of sign changes unless we knew how the terms were defined. The sequence -2, 3, -4, 5, is just an example. It’s neither the actual sequence nor does it indicate how the actual sequence will behave (i.e., don’t expect the actual sequence to alternate the signs just like the for-example sequence),.

To put it another way, if we only know statement one, how would you be able to tell whether the sequence is {-1, 1, -1} or {-1, 1, -1, 1}? The first one has two sign changes, and the second one has three sign changes.

If we only know that n is odd, in other words, if we only know that there is an odd number of terms in the sequence, how can we tell whether the sequence is {3, 5, 4, 2, 7} or {-6, 2, 1}? (Remember that we don’t know anything about the sequence, we only know that it has an odd number of terms.) In the first example, there are no sign changes, but in the second example, there is one sign change.

It is only when we know both statements that we know for sure that there is an even number of sign changes.
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I asked just because the question itself was just concerned with whether seq. has even or odd number of sign changes it never asked anything about sequence and its values.This confused me.

Hi All,

We're told that in any sequence of N NON-ZERO numbers, a pair of consecutive terms with opposite signs represents a sign change (for example, the sequence -2, 3, -4, 5 has three sign changes). We're asked if the sequence of nonzero numbers s1, s2, s3, . . . , sn has an EVEN number of sign changes. This is a YES/NO question. This question can be solved with a bit of Arithmetic and TESTing VALUES.

(1) sk = (-1)^k for all positive integers k from 1 to N.

While the information in Fact 1 might look complex, it actually defines how the sequence 'works.' The initial terms - and the pattern behind the sequence - are...
1st term = (-1)^1 = -1
2nd term = (-1)^2 = +1
3rd term = (-1)^3 = -1
4th term = (-1)^4 = +1
Etc.
We know how the sequence repeats, but we DON'T know how many terms are in it, so we don't know exactly how many sign changes there are (and thus, we don't know if the total number of changes is even OR odd).

IF....
There are 3 terms, then we have -1, +1, -1 and two sign changes - so the answer to the question is YES.
There are 4 terms, then we have -1, +1, -1, +1 and three sign changes - so the answer to the question is NO.
Fact 1 is INSUFFICIENT

(2) N is ODD.

Fact 2 tells us the number of terms in the sequence, but we know nothing about any of their values, so we don't know how many sign changes occur.
Fact 2 is INSUFFICIENT

Combined, we know...
sk = (-1)^k for all positive integers k from 1 to N.
N is ODD.

We know exactly how the sequence 'works' and that there are an ODD number of terms.
IF
There are 3 terms, then we have -1, +1, -1 and two sign changes - so the answer to the question is YES.
There are 5 terms, then we have -1, +1, -1, +1, -1 and four sign changes - so the answer to the question is YES.
There are 7 terms, then we have -1, +1, -1, +1, -1, +1, -1 and six sign changes - so the answer to the question is YES.
Etc.
Whenever there are an odd number of terms, the answer to the question is YES.
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Hi Experts - chetan2u nick1816 GMATBusters

Is the trick in this question NOT the assume that the set of non zero integers is necessarily consecutive ?

Just because the example given in the problem happened to be consecutive -- i thought obviously, the set is assumed to be consecutive

Hence i selected B

Hi jabhatta,

There are a couple of aspects to this prompt that are specifically-worded to help you answer the question that is ASKED.

First, the prompt opens with the phrase "In ANY sequences of N non-zero numbers...." meaning that it does NOT matter whether the numbers in the sequence are consecutive or not, as long as there are NO zeroes in the sequence.

Second, the example given (-2, 3, -4, 5) is merely there to give you an example of how to answer the given question (re: are there an EVEN number of 'sign changes' in the sequence?") - but it's not necessarily the sequence that we have to think about - and it is NOT a sequence of consecutive numbers. On a Number Line, CONSECUTIVE INTEGERS are "next to" one another (re: -2, -1, 0, 1, 2, 3, 4, 5). While the ABSOLUTE VALUES of -2 and +3 are "consecutive", the actual numbers -2 and +3 are NOT.

One of the 'keys' to any DS question is to consider what is possible and what is not based on the information that you're given. In this prompt, the only time we learn what the specific terms in the sequence are is in Fact 1 (but even that information does not tell us the NUMBER of terms in the sequence, so it's not enough information to answer the question that is asked).

GMAT assassins aren't born, they're made,
Rich

what if n=1? Zero sign change can not be considered neither even nor odd.. n>1 should'nt be the condition here for answer to be"C"?

If there are 0 sign changes, then the number of sign changes is still even because 0 is even

To clear up any confusion:

0 is even
0 is neither positive nor negative

Posted from my mobile device
_________________

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Answer: Option C

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chetan2u Bunuel

I have one query in this question, how do I know the sequence has more than one term.

Thanks!

Hi anupam87,

Based on the way the prompt is worded - especially the description of the sequence - it's highly likely that the sequence has at least 3 terms in it. That having been said, even if there is just 1 term in the sequence, the answer to the question remains the same (since 1 terms means that there would be 0 sign changes - and 0 is an even number).

GMAT assassins aren't born, they're made,
Rich

The question asks if the number of sing changes is even. I chose B because if S is positive there are 0 sign changes = 0 Even
if S is negative and n is odd then = Even number of sign changes.

so why it is not B?

-1,1,3……Only 1 sign change
-1,2(-1*-2),-4(2*-2)…..So 2 sign changes.

Unless you know something about the sequence, you cannot answer the question
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Doesn’t it tell you that the number of sign changes is even if the number of numbers is odd. -1,2,-3 = 2, and if you have 5 that is also even…?