GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 07 Dec 2019, 14:08 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # The sequence a1, a2, a3, ..., an of n integers is such that ak = k if

Author Message
TAGS:

### Hide Tags

Manager  Joined: 20 Feb 2007
Posts: 204
The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

3
50 00:00

Difficulty:   45% (medium)

Question Stats: 66% (01:43) correct 34% (02:04) wrong based on 556 sessions

### HideShow timer Statistics

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

Originally posted by Summer3 on 18 Mar 2007, 16:41.
Last edited by Bunuel on 11 Jun 2019, 04:29, edited 3 times in total.
Edited the question and added the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
The sequence a1, a2, a3, ..., an of n integers is such that  [#permalink]

### Show Tags

15
11
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?

We have following sequence:
$$a_1=1$$;
$$a_2=-a_1=-1$$;
$$a_3=3$$;
$$a_4=-a_3=-3$$;
$$a_5=5$$;
$$a_6=-a_5=-5$$;
...

Basically we have a sequence of positive and negative odd integers: 1, -1, 3, -3, 5, -5, 7., -7, 9, -9, ...

Notice than if the number of terms in the sequence (n) is odd then the sum of the terms will be positive, for example if $$n=3$$ then $$a_1+a_2+a_3=1+(-1)+3=3$$, but if the number of terms in the sequence (n) is even then the sum of the terms will be zero, for example if $$n=4$$ then $$a_1+a_2+a_3+a_4=1+(-1)+3+(-3)=0$$. Also notice that odd terms are positive and even terms are negative.

(1) $$n$$ is odd --> as discussed the sum is positive. Sufficient.
(2) $$a_n$$ is positive --> n is odd, so the same as above. Sufficient.

Hope it's clear.
_________________
VP  Joined: 01 May 2006
Posts: 1476
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

5
4
(D) for me Let us describe the few first terms to have a better idea of it works:
o a(1) = 1
o a(2) = -1
o a(3) = 3
o a(4) = -3

So,
o If n is even, then the sum of a(k) terms give 0. We always have couples of opposite number in the sequence.
o If n is odd, then the sum will be equal to n. All other numbers are in couple (negative/positive), giving 0 if we add them.

From 1
n is odd. Bingo, the sum is positive.

SUFF.

From 2
a(n) > 0... Then n must be an odd. Bingo, the sum is positive.

SUFF.
##### General Discussion
Intern  Joined: 26 Oct 2010
Posts: 19
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

1
I'll take a shot at explaining ...

If k is odd we know all the values are +ve and are equal to k (e.g. 1,3,5,7...)
If k is even we know all the values are -ve and are equal to the value of the prior term (e.g. a2 = -1,a4=-3... so the values will be, -1,-3,-5,-7 ....)

1, -1, 3, -3, 5, -5 .....

So as you can see at this point we know that for every value of k (when odd) we have a -ve value from when K is even, unless N (total terms) is odd in which case we will have one extra +ve term that will not cancel out. So if we have even number terms we know the result will be 0 (which is not positive). Therefore to get a positive sum we need one extra odd term.

Try it out,

N=5
1, -1, 3, -3, 5 (if add them, everything cancels out except 5, which is positive).

N=6
1, -1, 3, -3, 5, -5 (if add them, everything cancels out, result is not positive).

So, before looking at the statements we are able to rephrase the question to: "Is the number terms in the sequence odd?"

Statement 1: Gives us exactly that, therefore sufficient.
Statement 2: Well, it gives the same thing but instead of saying the number of terms is odd, it says the last term is +ve, which means the same thing as per our sequence above, there sufficient.

I hope this helps and makes sense.
Intern  Joined: 31 Oct 2012
Posts: 21
Re: The sequence a1, a2, a3, ..., an of n integers is such that  [#permalink]

### Show Tags

Now this is what i dont understand. They just mention "k" (i guess constant) but k can take any value -1,-3 or +2. SHouldnt the answer be B then because the statement 2 specifically says that an is positive.
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: The sequence a1, a2, a3, ..., an of n integers is such that  [#permalink]

### Show Tags

2
Now this is what i dont understand. They just mention "k" (i guess constant) but k can take any value -1,-3 or +2. SHouldnt the answer be B then because the statement 2 specifically says that an is positive.

$$k$$ in $$a_k$$ is a subscript, meaning that $$a_k$$ is $$k_{th}$$ term in the given sequence which starts from $$a_1$$, thus k must be some positive integer.

Complete solution is here: the-sequence-a1-a2-a3-an-of-n-integers-is-such-that-76926.html#p1162192

Hope it helps.
_________________
MBA Section Director V
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 7299
City: Pune
Re: The sequence a1, a2, a3, ..., an of n integers is such that  [#permalink]

### Show Tags

3
Now this is what i dont understand. They just mention "k" (i guess constant) but k can take any value -1,-3 or +2. SHouldnt the answer be B then because the statement 2 specifically says that an is positive.

dont go into complex things. Just visualize the sequence

It can be 2 way

1,-1, 3,-3, 5,-5, 7,-7 ending in negative term The sum will be zero in this case

1,-1, 3,-3, 5,-5, 7 ending in positive term The sum will be the last term of sequence

we have asked is the sum positive ? ----------> is the sequence as per 2nd case ? ----------> is the a(n) odd ? or is the a(n) positive ? both the statements answer these questions so both are sufficient
_________________
2020 MBA Applicants: Introduce Yourself Here!

MBA Video Series - Video answers to specific components and questions about MBA applications.

2020 MBA Deadlines, Essay Questions and Analysis of all top MBA programs
Director  G
Joined: 26 Oct 2016
Posts: 600
Location: United States
Schools: HBS '19
GMAT 1: 770 Q51 V44 GPA: 4
WE: Education (Education)
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

1
(1) n is odd
If n is 5, for instance, then a5=5, a4=-3, a3=3, a2=-1, a1=1 (I used each of the equations listed above to get these numbers I just followed the pattern) When added together [5+(-3)+(3)+(-1)+(1)] the result is +ve5
You could also plug-in 7 for n to make sure and you would still get a +ve result-follow the same pattern.
This statement alone is sufficient.

(2) an is +ve/ means the result must be +ve not zero or -ve

If n=5 (you get the same result as in statement 1)
However, if you let n=4/ following the same procedure as above (statement 1) you will see that the result is 0; therefore, one can conclude that n must be an odd number to get a +ve result. n cannot be even. This statement alone is also sufficient.

_________________
Thanks & Regards,
Anaira Mitch
Manager  B
Joined: 31 Dec 2017
Posts: 88
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

The sequence a1, a2, a3, ... an of n integers is such that ak=k if k is odd, and ak=−a(k-1) if k is even. Is the sum of the terms in the sequence positive?

(1) n is odd.
-->n = a(n)
For example:
a1 = 1
a2 = -1
a3 = 3
--->Sum of a1; a2; a3 = 3--->a positive integer-->The sum of odd terms is positive--->Sufficient

(2) a(n) is positive
--->n is odd--->The total number is odd--->The sum of odd terms is positive--->Sufficient
Intern  Joined: 25 Jan 2017
Posts: 3
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

What i dont understand is the assumption that the sequence starts with a positive term i.e. a1 is positive. If i start with a1= -1, a2= 1, a3 = -3 then the sum is negative. If i start with a1 = 1 the sum is positive. And if i take the values to be even, the sum is zero.

From statement 1, we know that n is odd. Therefore the sum is not 0, but it could be positive or negative. INSUFFICIENT.

From statement 2, we know An is positive, meaning there are odd terms in the sequence, with the last term being positive. SUFFICIENT.

So B should be the answer. Please tell me where i'm wrong in this solution.
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

beenish12 wrote:
What i dont understand is the assumption that the sequence starts with a positive term i.e. a1 is positive. If i start with a1= -1, a2= 1, a3 = -3 then the sum is negative. If i start with a1 = 1 the sum is positive. And if i take the values to be even, the sum is zero.

From statement 1, we know that n is odd. Therefore the sum is not 0, but it could be positive or negative. INSUFFICIENT.

From statement 2, we know An is positive, meaning there are odd terms in the sequence, with the last term being positive. SUFFICIENT.

So B should be the answer. Please tell me where i'm wrong in this solution.

We are given that $$a_k=k$$, so $$a_1=1$$ only.
_________________
Intern  Joined: 25 Jan 2017
Posts: 3
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

Couldnt k itself be negative?
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

1
beenish12 wrote:
Couldnt k itself be negative?

k there is an index number, a subscript indicating the position of a number in the sequence. There cannot be a sequence member with a position of -1. Also, we are given the the sequence is a1, a2, a3, ..., an, .... So, it starts with a1.
_________________
Intern  Joined: 25 Jan 2017
Posts: 3
Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  [#permalink]

### Show Tags

Thank you! I think i've finally got it. Re: The sequence a1, a2, a3, ..., an of n integers is such that ak = k if   [#permalink] 12 Nov 2019, 06:16
Display posts from previous: Sort by

# The sequence a1, a2, a3, ..., an of n integers is such that ak = k if  