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Sequence a1, a2, a3....an of n integers is such that ak = k

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Sequence a1, a2, a3....an of n integers is such that ak = k [#permalink] New post 18 Mar 2007, 15:41
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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) an is positive
[Reveal] Spoiler: OA

Last edited by Bunuel on 19 May 2012, 03:55, edited 1 time in total.
Edited the question and added the OA
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 [#permalink] New post 18 Mar 2007, 16:00
(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.
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Re: Sequences - GMATPrep [#permalink] New post 31 Aug 2007, 11:51
jp888 wrote:
The sequence a1, a2, a3,...an of n integers is such that ak = k if k is odd, and ak = -ak-1 if k is even. Is the sum of the terms in the sequence positive?

1) n is odd
2) an is positive

*note, figures after 'a' are in subscript, e.g. a1 and ak-1


this decomposes to 1 + (-1) + 3 + (-3) + 5 + (-5) + .....

so if n is odd all will cancel except the last positive number.
So 1) is sufficient

if an is positive, that is the only one remaining because the rest all pairs cancel out. So 2) is sufficent too...

Both are individually sufficient
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Re: Sequences - GMATPrep [#permalink] New post 29 May 2011, 10:47
carpeD wrote:
jp888 wrote:
The sequence a1, a2, a3,...an of n integers is such that ak = k if k is odd, and ak = -ak-1 if k is even. Is the sum of the terms in the sequence positive?

1) n is odd
2) an is positive

*note, figures after 'a' are in subscript, e.g. a1 and ak-1


this decomposes to 1 + (-1) + 3 + (-3) + 5 + (-5) + .....

so if n is odd all will cancel except the last positive number.
So 1) is sufficient

if an is positive, that is the only one remaining because the rest all pairs cancel out. So 2) is sufficent too...

Both are individually sufficient


Hi,
Could you please explain to me the decomposition part " this decomposes to 1 + (-1) + 3 + (-3) + 5 + (-5) + ....."

If ak=k when odd, then a1= 1 and ak=-ak-1 when even then a2 should be -2-1 =-3 . Therefore the sequence should be 1, -3, 3, -5, 5.... ?? not sure what im missing. please help.
Thanks
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Re: Sequences - GMATPrep [#permalink] New post 29 May 2011, 11:24
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.

Answer D.

I hope this helps and makes sense.
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Re: Sequences - GMATPrep [#permalink] New post 29 May 2011, 12:23
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Carol680 wrote:
Hi,
Could you please explain to me the decomposition part " this decomposes to 1 + (-1) + 3 + (-3) + 5 + (-5) + ....."

If ak=k when odd, then a1= 1 and ak=-ak-1 when even then a2 should be -2-1 =-3 . Therefore the sequence should be 1, -3, 3, -5, 5.... ?? not sure what im missing. please help.
Thanks


A_{1}=1
A_{2}=-A_{1}=-1

A_{3}=3
A_{4}=-A_{3}=-3

A_{5}=5
A_{6}=-A_{5}=-5

A_{7}=7
A_{8}=-A_{7}=-7
.
.
.

What do we see here:
A_1+A_2=1-1=0
A_3+A_4=3-3=0
A_5+A_6=5-5=0
...

Thus, if we have even number of elements in the series, their addition will always result in 0.
If we have odd number of elements, their addition will always result in +ve.

Q: Is there odd number of elements?

1. n is odd.
Precisely what we wanted to know.
Sufficient.

2. A_n is positive.
Means, the last element in the series is +ve.
We know, only odd number(index) has +ve values. All even values have -ve value.
Thus, there are odd number of elements.
Sufficient.

Ans: "D"
**********************

By the way, even if the statement said;
1. n is even
2. A_n is -ve.

The answer would be "D" because we would definitely know that the sum of terms is not +ve. It's zero.
**********************************
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Re: Sequences - GMATPrep [#permalink] New post 29 Jul 2011, 02:01
fluke wrote:
Carol680 wrote:
Hi,
Could you please explain to me the decomposition part " this decomposes to 1 + (-1) + 3 + (-3) + 5 + (-5) + ....."

If ak=k when odd, then a1= 1 and ak=-ak-1 when even then a2 should be -2-1 =-3 . Therefore the sequence should be 1, -3, 3, -5, 5.... ?? not sure what im missing. please help.
Thanks


A_{1}=1
A_{2}=-A_{1}=-1

A_{3}=3
A_{4}=-A_{3}=-3

A_{5}=5
A_{6}=-A_{5}=-5

A_{7}=7
A_{8}=-A_{7}=-7
.
.
.

What do we see here:
A_1+A_2=1-1=0
A_3+A_4=3-3=0
A_5+A_6=5-5=0
...

Thus, if we have even number of elements in the series, their addition will always result in 0.
If we have odd number of elements, their addition will always result in +ve.

Q: Is there odd number of elements?

1. n is odd.
Precisely what we wanted to know.
Sufficient.

2. A_n is positive.
Means, the last element in the series is +ve.
We know, only odd number(index) has +ve values. All even values have -ve value.
Thus, there are odd number of elements.
Sufficient.

Ans: "D"
**********************

By the way, even if the statement said;
1. n is even
2. A_n is -ve.

The answer would be "D" because we would definitely know that the sum of terms is not +ve. It's zero.
**********************************



Hi Fluke,

How will the ans be D here ?
By the way, even if the statement said;
1. n is even
2. A_n is -ve.


The answer would be "D" because we would definitely know that the sum of terms is not +ve. It's zero.



If n is even then all the terms cancel out so the sum of terms in the sequence is neither positive nor negative...so I is insufficient ..correct?? or its sufficient since we can definitely answer yes or no ???


What about statement 2 : If an is -ve then the sum of terms is also 0 here so same as case I ....it should be insufficient ...correct??


or the logic here is that since we can definitely answer both the statements its D.... PLease let me know...
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Re: Sequences - GMATPrep [#permalink] New post 29 Jul 2011, 02:52
siddhans wrote:
fluke wrote:
Carol680 wrote:
Hi,
Could you please explain to me the decomposition part " this decomposes to 1 + (-1) + 3 + (-3) + 5 + (-5) + ....."

If ak=k when odd, then a1= 1 and ak=-ak-1 when even then a2 should be -2-1 =-3 . Therefore the sequence should be 1, -3, 3, -5, 5.... ?? not sure what im missing. please help.
Thanks


A_{1}=1
A_{2}=-A_{1}=-1

A_{3}=3
A_{4}=-A_{3}=-3

A_{5}=5
A_{6}=-A_{5}=-5

A_{7}=7
A_{8}=-A_{7}=-7
.
.
.

What do we see here:
A_1+A_2=1-1=0
A_3+A_4=3-3=0
A_5+A_6=5-5=0
...

Thus, if we have even number of elements in the series, their addition will always result in 0.
If we have odd number of elements, their addition will always result in +ve.

Q: Is there odd number of elements?

1. n is odd.
Precisely what we wanted to know.
Sufficient.

2. A_n is positive.
Means, the last element in the series is +ve.
We know, only odd number(index) has +ve values. All even values have -ve value.
Thus, there are odd number of elements.
Sufficient.

Ans: "D"
**********************

By the way, even if the statement said;
1. n is even
2. A_n is -ve.

The answer would be "D" because we would definitely know that the sum of terms is not +ve. It's zero.
**********************************



Hi Fluke,

How will the ans be D here ?
By the way, even if the statement said;
1. n is even
2. A_n is -ve.


The answer would be "D" because we would definitely know that the sum of terms is not +ve. It's zero.



If n is even then all the terms cancel out so the sum of terms in the sequence is neither positive nor negative...so I is insufficient ..correct?? or its sufficient since we can definitely answer yes or no ???


What about statement 2 : If an is -ve then the sum of terms is also 0 here so same as case I ....it should be insufficient ...correct??


or the logic here is that since we can definitely answer both the statements its D.... PLease let me know...


Actually my bad,

2. A_n is 0 {Note: A_n can't be negative.}
Q: Is there odd number of elements?
A: No. Because Sum=0; number of elements must be even.
Sufficient.

1. n is even.
Q: Is there odd number of elements?
A: No. We are given the answer here.
Sufficient.
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Re: Sequences - GMATPrep [#permalink] New post 18 May 2012, 01:19
why do everyone assume that a1 is positive?
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Re: Sequences - GMATPrep [#permalink] New post 18 May 2012, 01:48
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mila84 wrote:
why do everyone assume that a1 is positive?


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;
...

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: Sequence a1, a2, a3....an of n integers is such that ak = k [#permalink] New post 18 May 2012, 20:47
can someone explain -

Should the question read here n consecutive integers...

Because for eg if we take -5, -3, -2 then sum = (-5) +(-3)+1 = -7

or if we take 4,5 then = -3+5 is positive..
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k [#permalink] New post 19 May 2012, 01:45
The way the sequence has been defined, a(k) + a(k+1) will always be 0 for every odd integer k.

Stt 1: If n is odd, that means the last term in the series is odd. As the sum of all preceding terms has to be zero, and a(k) is always positive when k is odd, the sum is always positive. Sufficient.

Stt 2: If a(n) is +ve, this means n is odd. By the same logic, the sum is always positive. Sufficient.

D it is.

@agdimple33: Yes, it should be n consecutive integers.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k [#permalink] New post 19 May 2012, 04:06
Expert's post
agdimple333 wrote:
can someone explain -

Should the question read here n consecutive integers...

Because for eg if we take -5, -3, -2 then sum = (-5) +(-3)+1 = -7

or if we take 4,5 then = -3+5 is positive..


Notice that n is a number of terms in the given sequence. I merged two topics so check out this solution: the-sequence-a1-a2-a3-an-of-n-integers-is-such-that-51526.html#p1086763

Hope it helps.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k [#permalink] New post 26 Apr 2014, 07:12
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k   [#permalink] 26 Apr 2014, 07:12
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