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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]

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Updated on: 18 May 2016, 22:06
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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

Originally posted by Summer3 on 18 Mar 2007, 15:41.
Last edited by Bunuel on 18 May 2016, 22:06, edited 2 times in total.
Edited the question and added the OA
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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18 May 2012, 01:48
10
9
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]

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18 Mar 2007, 16:00
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.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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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: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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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: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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29 May 2011, 11:24
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.

Answer D.

I hope this helps and makes sense.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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29 May 2011, 12:23
2
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: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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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: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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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: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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18 May 2012, 01:19
1
why do everyone assume that a1 is positive?
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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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]

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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]

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19 May 2012, 04:06
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]

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03 Mar 2015, 03:38
Can we not assume that the sequence begins with a negative number?
I initially thought a1 refers to any number(positive or negative).
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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03 Mar 2015, 04:39
1
CEZZAR89 wrote:
Can we not assume that the sequence begins with a negative number?
I initially thought a1 refers to any number(positive or negative).

No. We are given that $$a_k=k$$ and that the first term of the sequence is $$a_1$$, hence $$a_1=1$$.

Hope it's clear.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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19 May 2016, 00:25
Straight forward!

Given, a(n) = n if n = odd => a(1) = 1, a(3) = 3, etc.
Also, a(n) = -a(n-1) if n = even => a(2) = -a(1), a(4) = -a(3).

Therefore, the series is 1, -1, 3, -3, 5, -5, etc...
If we can answer the question, if we know the nth term. n = odd => nth term
n = even => 0

(a) n is odd.
Sufficient to answer.

(b) a(n) is positive.
Implies n is even . Suff.

Therefore, ans D.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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29 Jan 2017, 02:47
(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.

The answer is D.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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23 Apr 2017, 13:53
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.
**********************************
]]

Hello, Fluke (and everyone else).

Can you guys help me out with a question about this question? I understand the rationale used, except for one thing: Could we conclude that the fist statement (N is odd) be an odd negative number? > For example: n=-5

Thanks.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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02 May 2018, 16:45
Gabrielantonioreis wrote:
Hello, Fluke (and everyone else).

Can you guys help me out with a question about this question? I understand the rationale used, except for one thing: Could we conclude that the fist statement (N is odd) be an odd negative number? > For example: n=-5

Thanks.

Please note that the sequence of numbers are "a(1), a(2), a(3), ... a(n)" and question is "Is the sum of the terms in the sequence positive?". So, n cannot be -ve in this question.
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Re: Sequence a1, a2, a3....an of n integers is such that ak = k  [#permalink]

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14 Sep 2018, 18:54
For statement 1, how do we know the sequence starts at a1 instead of a2? If it starts at a1 and # of terms is odd, then it is positive. If it starts at a2, then you sum a2, a3 and a4 which results in a negative
Re: Sequence a1, a2, a3....an of n integers is such that ak = k &nbs [#permalink] 14 Sep 2018, 18:54
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