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The sequence a1, a2, a3, ..., an of n integers is such that

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The sequence a1, a2, a3, ..., an of n integers is such that [#permalink] New post 23 Mar 2009, 10:17
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 29 Dec 2012, 03:29, edited 1 time in total.
Renamed the topic, edited the question and added OA.
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Re: DS GMAT perp sequence [#permalink] New post 23 Mar 2009, 21:48
My Answer:
D. EACH statement ALONE is sufficient.

Explanation:

As,
ak=k if k is odd
Hence, a1=1
a3=3, a5=5 etc.

And as
ak=-ak-1 if k is even
Hence, a2=-a1=-1
a4=-a3=-3
a6=-a5=-5 etc.

So, if n is even,
a1+a2+....+an = a1+(-a1)+a3+(-a3)+....+an-1+(-an-1) = 0
All terms get canceled.

And if n is odd,
a1+a2+....+an = a1+(-a1)+a3+(-a3)+....+an-2+(-an-2)+an = an
As, ak=k if k is odd, an = n if n is odd
Only an remains at the end, which is a positive number equal to n.

Hence each statement satisfies it individually.

Now tell me the OA pls.
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink] New post 28 Dec 2012, 06:34
Can you explain this statement a little further, I just do not understand how you arrived at that statement Hence, a2=-a1=-1. Regards
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink] New post 29 Dec 2012, 03:30
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KevinBrink wrote:
Can you explain this statement a little further, I just do not understand how you arrived at that statement Hence, a2=-a1=-1. Regards


Stem says that a_k=k if k is odd. So, for k=1=odd we have that a_1=1.

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.

Answer: D.

Hope it's clear.
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink] New post 30 Dec 2012, 04:13
Accountant wrote:
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


In a sequence it always helps to observer a first few terms. Given these definitions : a1=1, a2=-a1 = -1, a3 = 3, a4=-a3 = -3

so it is clear consecutive terms from begining are canceling each other, i.e., 1-1+2-2+3-3 etc

Also, the sum is either positive in which case it is equal to the last odd term or it is zero.

So knowing either the term is odd or that last term was positive helps us know that sum of the terms are positive

D
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink] New post 03 Jan 2013, 21:38
a1 + (-a1) + a3 + (-a3) + .......

1. n is odd --> last term is an which is positive every other term cancels out
2. an can be +ve only if n is odd which will be the last term same as above
hence answer is D
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink] New post 14 Feb 2013, 04:55
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.
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink] New post 14 Feb 2013, 05:05
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maddyboiler wrote:
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.
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Re: The sequence a1, a2, a3, ..., an of n integers is such that [#permalink] New post 14 Feb 2013, 09:17
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maddyboiler wrote:
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
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink] New post 20 Nov 2013, 02:29
Bunuel wrote:
KevinBrink wrote:
Can you explain this statement a little further, I just do not understand how you arrived at that statement Hence, a2=-a1=-1. Regards


Stem says that a_k=k if k is odd. So, for k=1=odd we have that a_1=1.

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.

Answer: D.

Hope it's clear.


Thanks. But could a1 = -1? the question does not state that k>0 so I make a big mistake here... Please help to explain. Thanks a lot!
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink] New post 20 Nov 2013, 02:30
Thanks guys for the explanations of k. That really help me!!
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Re: The sequence a1, a2,a3,....an of n integers is such that [#permalink] New post 20 Nov 2013, 02:33
Expert's post
Cee0612 wrote:
Bunuel wrote:
KevinBrink wrote:
Can you explain this statement a little further, I just do not understand how you arrived at that statement Hence, a2=-a1=-1. Regards


Stem says that a_k=k if k is odd. So, for k=1=odd we have that a_1=1.

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.

Answer: D.

Hope it's clear.


Thanks. But could a1 = -1? the question does not state that k>0 so I make a big mistake here... Please help to explain. Thanks a lot!


We are told that a_k=k if k is odd. Now, substitute k=1 and see what you get.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: The sequence a1, a2,a3,....an of n integers is such that   [#permalink] 20 Nov 2013, 02:33
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